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1/ 



Elementary Mechanism: 



FOR 

STUDEiNTS OF MECHANICAL ENGLNEERING. 



ARTHUR T. WOODS, M.M.E., 

Late Professor of Mechanical Engineering, Washington University, 



AND 



ALBEKT ^. STAHL, M.E, 

Naval Con striictor, U S./Xavy ; Member Institution of Naval 

Architects ; Late Professor of Mechanical Engineering, 

Purdue University. 



FOUBTH EDITION, ENLARGED. 



%0 ^ ^B93 



NEW YORK : X^^O^^^^^^^?^/ \i 

D. VAN NOSTRAXD COMPANY, uIHQ^J 



33 Murray and 27 Waruen Street. 
1803. 






Copyright, 1885, 
By D. Van Nostband, 

Copyright, 1893, 
By D. Van Nostrand Company. 




0(o-?n2^7 



o 



PEEFAOE, 



Quite a number of treatises have appeared on the subject 
of Kinematics, or Pure Mechanism, most of which are now 
in print, so that a few words of explanation as to the reasons 
for publishing this book seem necessary. 

In searching for a text-book on this subject for the use 
of our classes of Mechanical Engineering students, we were 
unable to find a book which met our requirements. Some 
were so vague and incomplete as to be almost useless, while 
others were large, exhaustive treatises, more valuable as 
books of reference than as text-books for the use of stu- 
dents. The following pages were therefore prepared in the 
form of lectures ; the object being, to give a clear description 
of those mechanical movements which may be of practical 
use, together with the discussion of the principles upon 
which they depend. At the same time, all purely theoretical 
discussions were avoided, except where a direct practical 
result could be reached by their introduction. These lee- 
tures were used in our classes ; and, having proved com> 
paratively satisfactory in that shape, it was thought best to 
publish them, after making such improvements as our class- 
room experience dictated. 

iii 



IV PREFACE. 

We make little claim to originality of subject-matter, free 
use having been made of all available matter bearing on the 
subject. There is, in fact, very little room for such origi- 
nality, the ground having been almost completely covered 
by previous writers. Our claim to consideration is based 
almost entirely on the manner in which the subject has been 
presented. Accuracy, clearness, and conciseness are the 
points that we have tried to keep constantly in view. While 
much has been omitted that is of merely abstract interest, 
yet it is believed that nearly all that is of direct practical 
importance will be found in these pages. 

We have, in common with nearl}^ all other writers on 
this subject, closely followed tlie general plan of Professor 
Willis' "Principles of Mechanism." Other works which 
have been consulted and to which we are indebted are Ran- 
kine's " Machinery and Mill work," Reuleaux' " Le Construc- 
teur," and Goodeve's " Elements of Mechanism ; " and in a 
less degree, Belanger's " Cinematique," Reuleaux' "Kine- 
matics of Machinery," Robinson's "Teeth of Wheels," 
Grant's " Teeth of Gears," Appleton's " Cyclopaedia of Me- 
chanics," and Unwin's " Elements of Machine Design." 



PREFACE TO FOURTH EDITION. 



Adva:n^tage has been taken of the opportunity afforded 
by tlie necessity of publisliing another edition to correct 
all known typographical errors, to add a number of prob- 
lems, and to extend the work wherever such extensions 
seem desirable. It has been thought best to collect these 
additions in an Appendix rather than to interpolate them 
in the body of the book. The articles to which additions 
have been made have therefore been marked by a *, and 
the notes will be found in the Appendix under the cor- 
responding article numbers. 

In view of their own experience in teaching, the authors 
wish to emphasize the great importance, if not necessity, 
of carrying on mechanical drawing in direct connection 
with the class-room work. Mechanism is a practical sub- 
ject, which can be thoroughly taught only by actual and 
accurate graphic construction of the various means of 
transmitting and modifying motion, such as rolling curves, 
cams, link-work, and the teeth of wheels. 



CONTENTS. 



CHAPTER I. 

PAGE 

Introduction • 1 



CHAPTER 11. 

Elementary Propositions 10 

G-raphic Representation of Motion. — Composition and Resolution of Mo- 
tions. — Modes of transmitting Motion. — Velocity Ratio. — Directional 
Relation. 



CHAPTER III. 

Communication op Motion by Rolling Contact. — Velocity Ratio 

Constant. — Directional Relation Constant 27 

Cylinders. — Cones. — Hyperboloids. — Practical Applications. — Classifi- 
cation of Gearing. 



CHAPTER ly. 

Communication of Motion by Rolling Contact.— Velocity Ratio 

Varying. — Directional Relation Constant 52 

Logarithmic Spirals. —Ellipses. — Lobed Wheels. — Intermittent Motion. 
— Mangle Wheels. 



CHAPTER Y. 

Communication op Motion by Sliding Contact.— Velocity Ratio 
Constant. — Directional Relation Constant. — Teeth op 
Wheels 

Special Curves. — Rectification of Circular Arcs. — Construction of Special 
Curves. — Circular Pitch. —Diametral Pitch. 

vii 



YlU CONTENTS. 

CHAPTER yi. 

PAGE 

Communication of Motion et Sliding Contact. — Velocity Ratio 
Constant. — Directional Relation Constant. — Teeth of 

Wheels {Continued) 85 

Definitions. — Angle and Arc of Action. — Epicycloidal System. — luter- 
changeeble Wheels. — Annular Wheels. — Customary Dimensions. — 
Involute System. 

CHAPTER VII. 

Communication of Motion by Sliding Contact. — Velocity Ratio 
Constant. — Directional Relation Constant. — Teeth of 
Wheels {Continued) 113 

Approximate Forms of Teeth. — Willis' Method. — Willis' Odontograph. 
— G-rant's Odontograph. — Robinson's Odontograph. 



CHAPTER YIII. 

Communication of Motion by Sliding Contact. — Velocity Ratio 
Constant. — Directional Relation Constant. — Teeth of 

Wheels {Concluded) 131 

Pin Gearing. — Low-numbered Pinions. — Unsymmetrical Teeth. ~ Twisted 
Gearing. — Non-circular Wheels. — Bevel Gearing. — Skew-bevel Gear- 
ing. — Face Gearing. 



CHAPTER IX. 

Communication of Motion by Sliding Contact. — Velocity Ratio 

AND Directional Relation Constant or Varying 165 

Cams.— Endless Screw. — Slotted Link. — Whitworth's Quick Return 
Motion. — Oldham's Coupling. — Escapements. 



CHAPTER X. 

Communication of Motion by Linkwork. — Velocity Ratio and 

Directional Relation Constant or Varying 193 

Classification. —Discussion of Various Classes. — Quick Return Motion. — 
Hooke's Coupling. — Intermittent Linkwork. — Ratchet Wheels. 



CHAPTER XI. 

Communication op Motion by Wrapping Connectors.— Velocity 

Ratio Constant. — Directional Relation Constant .... 227 
Forms of Connectors and Pulleys. — Guide Pulleys. —Twisted Belts. — 
Length of Belts. 



CONTENTS. IX 

CHAPTER XII. 

PAGE 

Trains of Mechanism 240 

Value of a Train. — Directional Relation in Trains. — Clockwork. — Nota- 
tion. — Method of Designing Trains. — Approximate Numbers for 
Trains. 



CHAPTER XIII. 

Aggregate Combinations 263 

Differential Pulley. — Differential Screw Feed Motions. — Hpicyclic Trains. 
— Parallel Motions. — Trammel. — Oval Chuck. 



Appendix 289 

Problems 297 

Index 305 



ELEMENTARY MECHANISM. 



CHAPTER I. 



INTRODUCTION. 



1. A Machine is a combination of fixed and movable 
parts, interposed between the power and the work for the 
purpose of adapting the one to the other. 

This definition presupposes the existence of two things ; 
namely, a source of power, and a certain object to be ac- 
complished. The source of power may be one of the forces 
of nature applied directly, such as the expansive force of 
steam in a steam engine, or it may be obtained by the 
indirect application of such natural forces ; that is, the latter 
may have been already modified by some other machine. 
Thus, when a steam engine drives the machinery of a shop 
by means of a line of shafting, the latter may properly be 
considered as the source of power of the individual machine. 

2. Mechanism. — In designing a machine, we must take 
into consideration both the motions to be produced and the 
forces to be transmitted. But these two elements may most 
conveniently be discussed and investigated separately ; and 
such discussions and investigations constitute the two 
divisions of the general subject of mechanism ; namely, 
Pure Mechanism and Constructive Mechanism. 

1 



2 ELEMENTARY MECHANISM. 

Pure Mechanism, then, treats of the designing of ma- 
chines, as far as relates to the transmission and modification 
of motion, and explicitly excludes all considerations of force 
transmitted, or of strength and durability of parts. 

In order that the sense in which we shall use certain 
fundamental terms may be clearly understood, we shall now 
give an explanation of such words and phrases. 

3. Motion and Rest. — These terms are essentially 
relative. When a body changes its position with regard to 
some fixed point, it is said to be in motion relatively to that 
point ; when no such change is taking place, it is said to 
be at rest relatively to that point. Two bodies may evi- 
dently be in motion relatively to a third, and still be at rest 
with regard to each other. 

4. Path. — When a point moves from one position to 
another, it describes a line, either straight or curved, con- 
necting the two positions. This line is called its pai/i. But 
the path alone does not completely define the motion, for the 
point may move in the path in either of two directions; as, 
up or down, to the right or to the left, in the direction of the 
hands of a watch or the reverse. 

5. Kinds of Motion. — Motion may take place along 
either a straight or curved path ; in the former case it is 
termed rectilinear motion, and in the latter case curvilinear 
motion. In either case, when a moving point travels for- 
ward and backward over the same path, it is said to have a 
reciprocating motion. For example, the piston of a loco- 
motive has reciprocating rectilinear motion. In the par- 
ticular case where the reciprocating point moves in the arc 
of a circle, as, for example, the weight of a pendulum, it is 
said to oscillate^ or, by some, to vibrate. When the motion 
of a point is interrupted by certain definite intervals of rest, 
it is said to have an intermittent motion. The motion of the 
escape wheel of a clock is of this kind. 

6. Revolution and Rotation. — These terms are ordi- 



INTRODUCTION. 3 

Darily used syiionj^monsly, to denote the turning of a body 
about an axis ; and no aml^iguity is usually likely to arise 
from so using them. Thus, the fly wheel of an engine is 
said to rotate or revolve. By more strict definition, rotation 
should be applied only to the turning of a body about an 
axis which passes through it, while revolution is a more 
general term to include the motion of a body along a path 
which is a closed curve. Thus, the earth rotates about its 
axis and revolves about the sun. 

7. Velocity. — In addition to the path and direction of 
a moving body, there is another element necessary to com- 
pletely determine its motion, and that is its velocity. 

Velocity is measured by the relation between the distance 
passed over and the time occupied in traversing that distance. 
Velocity may be uniform and unchanging, or it may become 
greater or less ; and then changes may take place quickly or 
slowly, regularly or irregularly. But, for our purposes, it is 
sufficient to consider only two kinds of velocity, constant or 
uniform^ and variable. 

Velocity is expressed numerically by the number of units 
of distance passed over in one unit of time. The units of 
distance and time may be selected at pleasure ; but, for 
mechanical purposes, the most convenient units are feet and 
minutes ; and these will, in general, be employed throughout 
this volume. 

When a body moves with a uniform velocit}^, the distance 
passed over varies directly with the time. Thus, if by V 
we designate the velocity, and b}^ S the total distance passed 
over in the time T, we have S = VT. 

Again : if the velocity is given, we may find the time Tto 

traverse a given distance S, for T = — . 

When the distance and the time are given, we may deter- 
mine the velocity from the equation F = — . 



4 ELEMENTARY MECHANISM. 

For example, if a body moves at a uniform velocity over 
a distance of 100 feet, and occupies 5 minutes in doing so, 

it has a velocity V = — = = 20 feet per minute. 

In case the velocity is variable, these expressions do not 
give the velocity at any particular instant, but only the mean 
velocity for the whole time considered. The velocity at any 
particular instant is measured by the distance which the 
body would pass over in the next succeeding unit of time, 
were the velocity with which the body commences that unit to 
continue uniformly throughout it. Thus, if a railway train 
is slowing down in coming to a stop, its velocity is decreas- 
ing, but may, nevertheless, be measured at any instant. If, 
for instance, we say that the train has a velocity of 20 miles 
per hour, we mean, that, if it were to continue in motion for 
one hour at the velocity which it has at that instant, it 
would travel 20 miles. 

8. Ang-ular Velocity. — The most natural way of ex- 
pressing the velocity of a rotating body consists in stating 
the angle through which it turns, or the number of revolu- 
tions which it makes, in the unit of time. When the number 
of revolutions is given, it must usually be expressed as an 
angle before it can be used in calculation ; and the angle 
may be stated in degrees or in circular measure. For con- 
venience of comparison with linear velocities, we shall 
define angular velocity to be the velocity of a rotating bod}^ 
thus expressed in circular measure; i.e., as the quotient 
obtained by dividing the length of the arc subtending the 
angle through which it turns in one unit of time, by the 
length of the radius of that arc. All the points of a rotating 
body move with the same angular velocity, but the linear 
velocity of each point varies directly with its radial distance 
from the centre of motion. 

Let a = angular velocity of a body, R = radial distance 
of some point in that body, and V = linear velocity of that 



INTRODUCTION. 5 

point ; in other words, the length of the arc which it 
describes in the unit of time. 



V 
Then « = -r:- 

E 



Thus, if a locomotive having driving-wheels 5 feet in 
diameter is moving at a speed of 30 miles an hour, the 
linear velocity of a point on the rim of the wheel, relativel}^ 

to the frame of the enojine, is evidently F = — = — 

= 2640 feet per minute. The angular velocity of the wheel 

is therefore a = — = — -^ = 1056 feet per minute. 

The relation between the number of revolutions per 
minute and the angular velocity is readily found. Thus, 
let a wheel make N revolutions in T minutes. Let a point 
be taken at a radial distance E. Then this point will, in 
each complete revolution, describe a circle whose length is 
27rE ; in T minutes it will describe iVsuch circles, and travel 

a distance ^ttNE^ and its linear velocity F== -^ — . Hence 

its anovular velocity is a = — = = — . When T 

^ E TE T 

is unity, that is, when N is the number of revolutions per 
minute, a = 2??^, and N = —. Hence, in the above ex- 

27r 



ample of the locomotive driving-wheel, we find that the 

wheel makes N = — = = 168.07 revolutions 

27r 2 X 3.1416 

per minute. 

9. Periodic Motion. — During the operation of a ma- 
chine, it usually happens that the various moving parts go 
through a series of changes of motion which recur per- 
petually in the same order. The interval of time which 



6 ELEMENTARY MECHANISM. 

includes in itself one such complete series of changes is 
called a period^ and the character of the motion is described 
by the term periodic. The complete series of changes of 
motion included in one period is called a cycle. ' 

In periodic motion, the general law of the succession of 
changes is the same in successive periods, but the actual time 
may vary ; that is, the periods may be unequal in length. 
As a rule, however, the periods are equal, and the duration, 
magnitude, and law of succession of the changes are identi- 
cal, in successive periods ; such motion is known as uniform 
p)eriodic motion. 

lO. Classification of Parts of Machines. — As the 
work for which machines are designed varies so widely, and 
as they may be actuated by so many different kinds of power, 
we find great differences in them as to details of construction 
and manner of operation. But, in spite of these differences, 
every machine may be considered to consist of three classes 
of parts. At one end we have the parts which are specially 
designed to receive the action of the power; at the other 
we have those which are determined in form, position, and 
motion, by the nature of the work to be done. Between and 
connecting the former and the latter, we find the parts which 
are interposed for the purpose of transmitting and modifying 
the force and the motion ; so that, when the first parts move 
according to the law assigned them by the action of the 
power, the second must necessarily move according to the 
law required by the character of the work. These three 
classes of parts are so far independent of one another, that 
any kind of work may be done by any kind of power, and 
by means of various combinations of interposed mechanism. 
The motion of the parts which receive the action of the 
power must be transmitted to the working-parts ; and, as 
the action of the latter is usually very different from that of 
the former, it follows that the motion must be modified, 
during transmission, according to certain definite conditions. 



INTRODUCTION. 7 

This modification is accomplished by means of the interposed 
mechanism above mentioned, and it is to the discussion of 
the methods by which motion may be transmitted and modi- 
fied that the following pages are devoted. 

11. A Train of Mecliaiiisni is composed of a series of 
movable pieces, each of which is so connected with the 
frame-work of the machine, that when in motion every point 
of it is constrained to move in a certain path, in which, how- 
ever, if considered separately from the other pieces, it is at 
liberty to move in the two opposite directions, and with any 
velocity. Thus, wheels, pulleys, shafts, and rotating pieces 
generally, are so connected with the frame of the machine, 
that any given point is compelled, when in motion, to de- 
scribe a circle round the axis, and In a plane perpendicular 
to it. Sliding pieces are compelled by fixed guides to de- 
scribe straight lines, other pieces to move so that their points 
describe more complicated paths, and so on. 

12. These pieces are connected in successive order in 
various ways so that when the first piece in the series is moved 
from any external cause, it compels the second to move, 
which again gives motion to the third, and so on. The vari- 
ation in the laws of motion of the different pieces of a train 
is effected by the mode of connection. 

13. Modes of Connection. — One piece may transmit 
motion to another by direct contact^ or by means of an inter- 
mediate connector. In the latter mode of connection, the 
motion of the intermediate piece is usuall}^ of no importance, 
the object to be secured being simply the proper relative 
motion of the two primary pieces. Two pieces connected in 
either of the above ways, so that a definite motion of one of 
the pieces will produce an equally definite motion of the 
other, form an elementary combination. A train of mechan- 
ism evidently consists of a series of such elementary com- 
binations, each piece receiving motion from the one that 
precedes it, and transmitting motion to the one next in order. 



8 ELEMENTARY MECHANISM. 

14. That piece of an elementaiy combination to which 
motion is imparted from some exti-aneous source is termed 
the driver ; and that piece whose motion is received from, 
and governed by, the driver is termed the foUoiver. 

15. Velocity Katio and Directional Relation. — It 
has been ah^eady shown that the paths of the pieces in an 
elementary combination are fixed, and depend on the con- 
nection of the pieces with the framework of the machine ; 
while their velocity and direction of motion may vary, and 
must be determined for each instant of action. Thus, in 
comparing the motions of the pieces for successive instants, 
we may find changes of velocity or of direction, or both. 
But, while the absolute velocities and the absolute directions 
of both pieces may be liable to continual variation, it is 
evident that there will exist, at each instant, a certain 
definite ratio between the velocities, and an equally definite 
relation between the directions, of the driver and follower. 
This velocity ratio and this directional relation will depend 
solely on the manner in which the two pieces are connected, 
and will be entirely independent of their absolute velocities 
or directions. The velocity ratio, and also the directional 
relation, may be constant during the entire period, or either 
or both may vary. For example, if two circular wheels 
turning on fixed axes gear with each other, their velocity 
ratio is constant. If one wheel is twice as large as the 
other, it will make only one-half as many turns in the same 
time, or its angular velocity will be half that of the smaller 
wheel. But during any changes in velocity whatsoever, as 
one wheel cannot rotate without turning the other, and as the 
respective radii of contact do not change in length, the ratio 
of their velocities at any instant is the same ; that is, such 
wheels have a constant velocity ratio. And so, also, of the 
relative directions of the rotations. If the wheels are in 
external gear, they will turn in opposite directions ; if in 
internal gear, in the same direction : but in either case the 



INTRODUCTION. 9 

directional relation will remain constant, without regard to 
any change of absolute direction of the driver. 

If the two wheels are elliptical, however, as those shown 
in Fig. 42, the directional relation will be constant, while the 
velocity ratio will vary according to the varying lengths of 
the radii of contact. 

If, then, in addition to the paths of both driver and fol- 
lower, we have determined their velocity ratio, and the 
directional relation of their motion, for every instant of an 
entire period, our knowledge of the action of the combination 
will be complete. 



10 ELEMENTARY MECHANISM. 



CHAPTER II. 

ELEMENTARY PROPOSITIONS. 

Graphic Bepresentation of Motion. — Composition and Resolution 
of Motions. — Modes of Transmitting Motion. — Velocitij Ratio. 
— Directional Relation. 

16. Graphic Representation of Motion. — The prob- 
lems relating to the motions of points may be most readily 
solved by geometrical construction. It is evident that the 
rectilinear motion of a point may be represented by a straight 
line ; for the direction of the line may represent the direction 
of the motion, while the velocity may be indicated by its 
length. When a point moves in a curve, its direction of mo- 
tion at any instant is the same as the direction of the tangent 
to the curve at the point considered. Hence the curvilinear 
motion of a point may be represented in the same mannei 
as the rectilinear motion, using the direction of the tangent as 
the direction of the straight line above mentioned, and making 
its length proportional to the velocity, as before. 

By thus representing the motion of properly selected 
]:>oints, we may establish certain relations, by purely geomet- 
rical reasoning, which will not only enable us to obtain the 
velocity ratio and the directional relation m the particular 
phase represented, but will lead to, and almost involve, the 
accurate construction on paper of the movements considered. 
The latter is such an important advantage in practical work, 
that this method is greatly to be preferred, and has been 
adopted in this volume. 



ELEMENTARY PROPOSITIONS. 11 

17. Composition of Motions. — If a material point 
receives a single impulse in a given direction, it will move 
in that direction with a certain velocity ; and, as above 
explained, its motion may be represented by a straight line 
havinsf the same direction as the motion, and of a leno;th 
proportional to the velocity. If a point receives, at the 
same time, two impulses in different directions, it will obey 
both, and move in an intermediate direction with a velocity 
differing from that due to either impulse alone. Such a 
point may receive, at the same instant, any number of 
impulses, each one tending to impart to it a motion in a 
definite direction and with a certain velocity. Now, it is 
evident that the point can move only in one direction and 
with one velocity; this motion is called the resultant; and 
the separate motions which the different impulses, taken 
singly, tended to give it, are called the components. 

18. Parallelog^ram of Motions. — Given two com- 
ponent motions of a point, to find the resultant. 

In Fig. 1, let the point A be acted on at the same time 
by two impulses, tending to give it the motions represented, 
in direction and velocity, by the 
straight lines AB and AD respec- 
tively. Through B draw BC 
parallel to AD ; through D draw 
DC parallel to AB\ join AC. 
Then AC will represent, in direc- i"ig. i 

tion and velocity, the motion 

which the point A will have as the result of tlie two im- 
pulses which separately would have produced the motions 
AB and AD respectively. The length of the resultant may 
be altered by varying the lengths of the components or the 
angle between them, but in no case can it exceed their sum 
nor be less than their difference. 

This proposition is known as the parallelogram of motions^ 
and may be thus stated ; — 




12 



ELEMENTARY MECHANISM. 



If two component motions be represented, in direction and 
velocity, by the adjacent sides of a parallelogram, the 
resultant will be similarly represented by the diagonal 
passing through their point of intersection. 

19. Polyg-on of Motions By a repetition of the 

above process, we may find the resultant of any number of 
simultaneous independent components. 




Fig 

In Fig. 2, let AB, AD, AF, represent three such com- 
ponents. We first compound any two of them, as AB and 
AD, by completing the parallelogram ABCD, and find the 




:Fig. 



resultant AC. We next compound AC with AF in a similar 
manner, and find the resultant AE. The latter is evidently 
the resultant of the three components. 



ELEMENTARY PROPOSITIONS. 18 

This process may be continued for any number of com- 
ponents, and it makes no difference in what order they are 
taken. In Fig. 3, for instance, we have the same com- 
ponents as in Fig. 2, and find the same resultant, AE, 
though the composition is carried on in a different order. 

20. Resolution of Motion. — This is the inverse of the 
process just explained. It is obvious, that, if two or more 
independent motions can be compounded into a single 
equivalent motion or resultant, the latter can be again 
separated, or resolved, into its components. But it evidently 
makes no difference whether the single motion to be resolved 
is the resultant of a previous composition, or whether it is 
an original independent motion. Any single motion can be 
resolved into two others, each of these again into two others, 
and so on as far as desn^ed ; these components being given 
any directions at pleasure. In Fig. 4, let AC represent the 
given motion. Through A draw the indefinite lines AE and 
AHin the directions in which it is 
desired to resolve AC. Through 
C draw CB parallel to AH, and 
intersecting AE at B ; also CD 
parallel to AE, and intersecting 
AH at D. Then AB and AD 
will be the components required ; mig. 4! 
and it is evident that by their 

composition (Art. 18) we would find their resultant to be 
AC, the given motion. 

21. Communication of Motion by Direct Contact. 
— In Fig. 5, let AD and BC be two successive pieces of a 
train of mechanism, turning about the centres A and B 
respectively. Let AD be the driver, turning the follower 
BC, by contact, between the curved edges, as shown. Let 
c be the point of contact between the two pieces ; and let 
the driver move the follower, until they occupy the positions 
shown by dotted lines, the points a and b having come in 




14 



ELEMENTARY MECHANISM. 




Fig. 5 



contact at d. During this motion, every point of the curved 
edge of the follower between b and c has been in contact 
with some point of the curved edge of the driver between a 
and G. If be is not equal in length to ac, it is evident that 
sliding of one edge on the other must have 
taken place through a space equal to their 
difference ; but, if be = ac, there will have 
been no sliding. In the latter case the mo- 
tion is said to be communicated by rolling 
contact, and in the former case by sliding 
contact.* 

Motion, then, may be communicated by 
two kinds of direct contact : — 

1 . By rolling contact, when each point of 
contact of the driver with the follower is 
continually changed, but so that the curve 
joining any given pair of points of contact 
of the driver shall be equal in length to the 
curve joining the respective points of the follower. 

2. By sliding contact, when each point of contact of the 
driver with the follower is continually changed, but so that 
the curve joining any given pair of points of contact of 
the driver shall not be equal in length to the curve joining the 
respective points of the follower. 

In contact motions, one or both of the curves must be con- 
vex ; and, in the former case, the convex edge must have a 

* More strictly speaking, sliding contact should be defined as that 
motion in which every point of contact of one piece comes into contact 
with all the consecutive points, in their order, of a line in the other. 
Thus, the piston of a steam engine moves in true sliding contact with 
the interior surface of the cylinder. When this definition of sliding 
contact is adopted, it is usual to class under the head of mixed contact 
those contact motions which partake of both rolling and sliding. But, 
for our purposes, it is sufiicient to distinguish between contact which 
is rolling and that which is not; designating by the term " sliding " not 
only that which is purely so, as just defined, but also the cases just 
spoken of as mixed contact. 



ELEMENTARY PROPOSITIONS. 15 

sharper curvature than the concave edge. If this condition 
is not fulfilled, contact will take place at discontinuous 
points. 

22. Comniuiiicatioii of Motion by Interniecliate 
Connectors. — Such intermediate connectors maj^ be divided 
into two general classes : links, which are rigid, and must 
be jointed or pivoted to both the driver and follower ; and 
bands, or iDrapping connectors, which are flexible. 

The former class includes all forms of rigid connectors 
which can transmit motion by pushing or pulling, such as 
connecting-rods, locomotive side rods, etc. ; the latter in- 
cludes all forms of connectors which can transmit motion 
by pulling only, such as belts, ropes, chains, etc. 




3Fig 



In Fig. 6, let AP, BQ, be driver and follower, moving 
about the centres A and B respectively, and connected by 
the link PQ. If AP is turned so as to occupy another 
position, Ap or Ap^, it will, by means of the link, move the 
arm BQ into the position Bq, or Be/. If the driver pu>iJt 
tlie follower, the connector is necessarily rigid, and, as just 
stated, belongs to the general class of linlxs. But the con- 
nector may be flexible, as in Fig. 7, where ACE is the driver, 
and BDF the follower, turning about the centres A and B 
respectively, and connected by a flexible but inextensil)le 
band which lies in the direction of the common tangent of 
the two curves. If the driver be moved in the direction 
of the arrow, it will, by means of this connector, turn the 



16 



ELEMENTARY MECHANISM. 



follower as indicated 
C/ 




I^ig.7 



and the connector will unwrap itself 
from the curved edge of 
the latter, and wrap itself 
on that of the former. 
By means of this form of 
intermediate connector, 
which belongs to the 
general class of bands 
or ivrapping connectors^ 
it is evident that motion 
can be transmitted by 
pulling or tension only. 

23. Modes of Trans- 
mission of Motion. — 
Every elementary com- 
bination may be classi- 
fied according to one of 
the four modes of trans- 



mission of motion just defined ; namely, 



1. Rolling contact. 

2. Slidins: contact. 



3. Link work. 

4. Wrapping connectors. 



^i 24. Velocity Ratio in Linkwork. — In Figs. 8 and 
9, let AP^ BQ, be two arms, turning on fixed centres A and B 
respectively, and connected by the rigid link PQ. Since the 
arm AP turns about the centre A, the point P will move in 
the arc of a circle, and hence its direction of motion at any 
instant will be represented by the tangent to that arc ; that is, 
by a line perpendicular to the radius AP. Draw Pa per- 
pendicular to AP, and of such a length as to represent the 
velocity V of the point P in that direction at that instant. 
Similarly, draw Qb perpendicular to BQ to represent, by 
length and direction, the velocity v of the point Q at that 
instant. Let fall the perpendiculars AN and BM from the 
fixed centres of motion upon the line of the link ; and let T 
be the point of intersection of the line of the link with the 
line of centres. 



ELEMENTARY PROPOSITIONS. 



17 



The resolved velocity of P along the line of the link is 
Fcos aPc = Fcos PAN = Fcos ; and the resolved velocity 
of Q along the line of the link is v cos b Qd -- v cos QBM 
— V cos (/). Since the link PQ is rigid, and can be neither 



:>a 




Fig. 8 



Fig. 9 



extended nor compressed, the resolved velocities of the points 
P and Q along the line of the link must be equal. Hence, 
Fcos (9 = rcos^. (1) 

v_ _ cos^ /2\ 

F cos c^ 
Let a = angular velocity of P about the centre A ; and 
a = angular velocity of Q about the centre B ; 
then (Art. 8) , 

"^ PA' "^ QB' 



PA ^ 
QB V 



PA cos 
QB cos (/) 



AN 
BM' 



(3) 



18 



ELEMENTARY MECHANISM. 



Also, since angle ATN = angle BTM^ we have, 
a BM BT 



(4) 



Hence, in the communication of motion by linkwork, — 

1. The angular velocities of the arms are inversely pro- 
portional to the perpendiculars from the fixed centres of 
motion upon the line of the link. 

2. The angular velocities of the arms are inversely pro- 
portional to the segments into which the line of the link 
divides the line of centres. 

25. This proposition may also be proved by means of the 
instantaneous centre. 




In Figs. 10 and 11 the link PQ may be regarded as 
turning, during each instant of its motion, about some centre 
in space. This centre may be constantly changing its posi- 
tion in space, and also with regard to the line PQ itself ; 
but at any given instant every point in PQ has the same 



ELEMENTARY EROrOSITIONS. 



19 



angular velocit}^ about tliis centre, and moves in a direction 
perpendicular to the line joining it to the centre, and with a 
linear velocit}^ proportional to its distance from it. As P 
moves perpendicularly to AP^ the centre must lie in AP 




^'ig. 11 



(produced if necessary) ; and as Q moves perpendicularly 
to BQ^ it must also lie in BQ (produced if necessary) : 
hence it will be found at the intersection of these two lines 
at 0. Let V and v represent the linear velocities of P and 
Q respectively. As both P and Q have the same angular 
velocity about 0, their linear velocities will be proportional 
to their distance from that point ; that is, 

V : V '.'. PO : QO. 

Let a and a be the angular velocities of P and Q about A 
and B respectively. Then 



AP ' BQ 



PO . QO 
AP ' BQ' 



20 ELEMENTARY MECHANISM. 

Let fall the perpendiculars OR^ AN^ and BM upon the 
line of the link ; then, from the similar triangles ANP and 
ORP, BQMsxud OQR, BT3I and ATN, we have 

P^^OR ^^^^ QO^OR^ 

AP AN BQ BM 

Hence 

a' ^ OR^ AN ^ AN^ ^ AT 
a BM OR BM BT" 
as before. 

26. Directional Relation From Figs. 8, 9, 10, and 

11, it is evident that the directional relation of the rotations 
of the two arms depends on the position of the centres A 
and B with reference to the line of the link PQ. If the}^ are 
on the same side of PQ, the rotations will take place in the 
same direction ; if on opposite sides, the rotations will be in 
contrary directions. 

27. Velocity Ratio in Wrapping* Connectors. — In 
Figs. 12 and 13, let AG and BD be two curved pieces 




moving about the fixed centres A and B respectively, and 
let them be connected by the flexible but inextensible band 
EPQF, fastened to them at E and F. If AC be turned in 
the direction of the arrow, it will cause BD to turn by means 
of the band, which will unwrap itself from the curved edge 
of BD, and wrap itself on that of AC. Let P and Q be 
the points at which the line of the band is tangent to the 



ELEMENTARY PROPOSITIONS. 



21 



curved edges. These points must move perpendicularly 
to the radii AP and BQ ; and the action at any instant 
is precisely the same as that of two arms, AP and BQ, 
connected by a link, PQ, as disciussed in the preceding 
articles. Hence, letting fall the perpendiculars AN and BM 




E^.ig.13 



upon the common tangent, which is the line of the wrapping- 
connector, and finding the intersection T of the latter and 
the line of centres, it follows, that, in the communication of 
motion by wrapping connectors, — 

1 . The angular velocities of the pieces are inversely pro- 
portional to the perpendiculars from the fixed centres of 
motion upon the line of the wrapping connector. 

2. The angular velocities of the pieces are inversely pro- 
portional to the segments into which the line of the wrapping 
connector divides the line of centres. 

28. Directional Relation. — From Figs. 12 and 13, it 
is evident that the directional relation of the rotations of the 
two pieces depends on the position of the centres A and B 
with reference to the line of the wrapping connector PQ. 
If they are on the same side of PQ, the rotations will take 
place in the same direction ; if on opposite sides, the 
rotations will take place in contrary directions. 



22 



ELEMENTARY MECHANISM. 



29. Velocity Ratio in Contact Motions. — In Figs. 14 
and 15 let the sectioned portions in contact at p represent 
parts of two curved pieces turning about fixed centres A and 
B. These curved edges may both be convex, as in the fig- 
ures ; or one may be concave, provided that the curvature 
of the convex edge is sharper than that of tlie concave edge. 




^^.a- 



If the lower piece be turned in the direction of the arrow, 
it will drive the upper piece, and compel it to turn as 
indicated. 

Draw Hli^ the common normal, and Rr^ the common tan- 
gent, to the curves at the point of contact. 

The pointy of the lower piece moves at any instant in a 
direction perpendicular to the radius Ap. Draw pa perpen- 
dicular to ^p^and of such a length as to represent the 
velocity Fof the point p in that direction at that instant. 
Similarly, let ph drawn perpendicular to Bp represent the 
velocity v of the point p of the upper piece at that instant. 
Resolving these velocities along and perpendicular to the 
common normal Hh^ it is evident that the component along 



ELEMENTARY PROPOSITIONS. 



23 



the normal must be tliat wliich transmits motion from the 
driver to the follower ; for the component perpendicular to 
the normal, acting tangentiall}^ to the two curves, can evi- 
dently transmit no motion from the one to the other. 

In considering the communication of motion through a very 




Fig'. 15 



small angle, we may substitute for the curves, circular arcs 
having the same curvature as the curves themselves at the 
point of contact. The centres of these circular arcs will evi- 
dently lie at some points P and Q in the normal Hh. But 
the length PQ, being thus equal to the sum of the radii of two 
circles, will be constant during the small motion considered. 
Hence, joining AP and jBQ, we find that the angular motion 
of the tAvo contact curves, for that instant, will be exactly 
the same as that of two arms, AP and BQ^ connected by the 
link PQ. Hence, the relation of the angular velocities will 
be expressed in the same manner as in Art. 24. 

Letting a = angular velocity of lower piece about A^ and 
a = angular velocity of upper piece about B, 



24 ELEMENTARY MECHANISM. 

we have, as before, , ' 

a' ^ AN ^ AT 
a BM BT' 

Hence, in the communication of motion by contact, — 

1. The angular velocities of the pieces are inversely pro- 
portional to the perpendiculars from the fixed centres of 
motion upon the common normal. 

2. The angular velocities of the pieces are inversely propor- 
tional to the segments into which the common normal divides 
the line of centres. 

30. Directional Relation From Figs. 14 and 15, 

it is evident that the directional relation of the rotations 
depends on the position of the centres A and B with 
reference to the normal IIli. If they are on the same side 
of Hh^ tlie rotations will take place in the same direction ; if 
on opposite sides, the rotations will take place in contrary 
directions. 

31. Condition of Constant Velocity Ratio. — The 
value of the velocity ratio (Art. 29) is 



AT 
BT 



Now, in order that this expression shall have a constant 
value, the ratio of AT io BT mxi^i remain unchanged. But, 
as AT + BT = AB, which is itself constant, it follows, 
that, in order to preserve the constancy of the above ratio, 
the actual lengths of ^T and BT must not vary; in other 
words, the point T must remain fixed in position. Hence 
we see, that, in order to obtain a constant velocity ratio in 
contact motions, the curves must be such that their common 
normal at the point of contact shall always cut the line of 
centres at the same point. 



ELEMENTARY PROPOSITIONS. 25 

32. Condition of Rolling' Contact. — It has been 
stated (Art. 29) that the components of the velocities Fand 
r, Figs. 14 and 15, along the common normal Hh^ represent 
the transmitted motion. As pointed out by Professor Ran- 
kine, these components must be equal ; or Fcos^ = v cos <^, 
using the same notation as in Art. 24. If this equation were 
not true, either one curve would move away from the other, 
or else one would intersect the other, both of which are 
manifestly impossible while contact exists. 

As the components of V and v along the tangent Rr must 
represent the lost motion, it is evident that the velocity with 
which the curves are sliding over each other will be repre- 
sented by the difference between these tangential components 
in the case of Fig. 14, and by their sum in the case of Fig. 
15. In other words, the velocity of sliding is Fsin — v sin <^ 
in the former case, and Fsin^ + vsincfi in the latter. In 
order that there shall be pure rolling contact, it is evidently 
necessary that these expressions shall be equal to zero. 

Thus, we must have in Fig. 14, 

Fsin — V sincf> = 0. 
VsiuO = vsincfy. (1) 

But, as above stated, 

V cos = V cos cfi. (2) 

Dividing (1) by (2) we get 

tan e = tan </>, (3) 

which can only be true when = eft; in other words, when 
V coincides in direction with v. But Fand v are perpen- 
dicular to Ap and Bp respectively ; hence the latter must 
also coincide in direction, and as A and B are fixed, it fol- 
lows that p must lie in the line of centres AB. The condition 
of rolling contact, then, for curA^es revolving in the same 
plane about parallel axes, is that the point of contact shall 
always lie in the line of centres. 



26 ELEMENTARY MECHANISM. 

In most cases of sliding contact, the point of contact is not 
fixed in position, and the amount of sliding will vary with 
the distance of the point of contact from the line of centres. 
We have found (Art. 29) that the velocity ratio in contact 
motions depends on the position of the point of intersection 
of the common normal with the line of centres. But in roll- 
ing contact, the point of contact must lie on the line of cen- 
tres, and must hence be identical with the point of intersection 
just mentioned. Hence, in rolling contact, the angular ve- 
locities of the pieces are inversely proportional to the seg- 
ments into which the point of contact divides the line of 
centres. 

33. Similarity in all Modes of Transmission By 

comparing the deductions of Arts. 24-30, we find a great 
similarity between the various modes of transmitting motion, 
so far as the velocity ratio and the directional relation in the 
various cases are concerned. 

If we designate by line of action the line of the link in link- 
work, the line of the wrapping connector, and the common 
normal in contact motions, we may express the laws govern- 
ing the action of any elementary combination in which the 
pieces rotate about fixed parallel axes as follows : — 

1. The angular velocities are inversely proportional to the 
perpendiculars let fall from the centres of motion upon the 
line of action. 

2. The angular velocities are inversely proportional to 
the segments into which the line of action divides the line of 
centres. 

3. The rotations have the same direction if the centres 
of motion lie on the same side of the line of action, and 
contrary directions if they lie on opposite sides of that line. 



MOTION BY ROLLING CONTACT. 27 



CHAPTER III. 

COMMUNICATION OF MOTION BY ROLLING CONTACT. 

VELOCITY RATIO CONSTANT. 

DIRECTIONAL RELATION CONSTANT. 

Cylinders. — Cones. — Ilyperholoids. — Practical Applications. — 
Classification of Gearing. 

34. It has been shown (Art. 32) that, in the roUing con- 
tact of curved pieces revolving in the same plane about fixed 
parallel axes, the point of contact will always lie in the line of 
centres, and that the angular velocities are inversely propor- 
tional to the segments into which the point of contact divides 
that line. 

Therefore, if the velocitiy ratio of two such pieces in roll- 
ing contact is constant, these segments must be constant, 
and the curves must have a constant radius ; in other words, 
the curves must be circular arcs turning about their centres, 
and no other curves will satisfy the conditions. 

Axes Parallel . 

35. Rolling Cylinders. — In Fig. 16, let AC, BD, be 

parallel axes mounted in a framework, b}' which they are 
kept at a constant distance from each other. Let E and F 
be two cyhnders, fixed opposite to each other, one on each 
axis, and concentric with it ; the sum of their radii being 
equal to the distance between the axes. 

The cylinders will, therefore, be in contact in all positions, 



28 



ELEMENTARY MECHANISM. 



the line of contact being a common element of both. If one 
cylinder be made to rotate, it will drive the other b}^ rolling 
contact, and compel it to rotate. The linear velocity of 
every point in the cylindrical surface of either wheel must 
evidently be the same. 




IPig; 16 

Let R be the radius of the driver, and r the radius of the 
follower. Let the circumference of the driver be divided 
into iV equal parts, and let the circumference of the follower 
contain n of these parts. Let P and p be the periods or 
times of rotation ; L and I the number of rotations in a given 
time, or the synchronal rotations of driver and follower re- 
spectively ; and, as before, let a and a be their angular veloci- 
ties. Then 

^ = :5 = ^ = ?. = 1- 

a r n p L 



and it is evident that these ratios will hold, without regard to 
the absolute velocities. 

36. If the cylinders roll together by external contact, as 
in Fig. IG, they will evidently rotate in opposite directions. 



MOTION BY ROLLING CONTACT. 



29 



If it is desired to have them rotate in the same direction, one 
wheel is given the form of an annulus, or ring, as in Fig. 17, 
to which the other wheel is tangent internally. The rolling 
surfaces are cylinders, as before ; the line of contact is an 
element of both cylinders ; and the relations stated in the last 
article are equally true for this case, the only change being, 
that the rotations now take place in the same direction. The 
difference of the radii is evidently the distances between cen- 
tres. Thus, if we have given the distance between two axes, 
and the velocity ratio of driver and follower, expressed in 
any of the above terms, we can readily find the radii of 
wheels which will answer the given conditions. 




Fig. XT' 

If the axes of rotation are not parallel, they may or may 
not intersect ; and these cases will he considered separately. 

Axes Intersecting. 



37. Rolling Cones. — The conclusions arrived at in 
Art. 34 follow directly from our propositions concerning 
rolling contact ; the circles in contact being in the same 
plane, and rotating about fixed parallel axes. A little con- 
sideration will, however, make it clear, that, if the axes be 



30 



ELEMENTARY MECHANISM. 



turned in their common plane about the point of contact of 
the two circles, the latter will, at any angle, have a common 
tangent at this point. This tangent will be the line of in- 
tersection of the planes in which the two circles lie. Both 
circles will be in true rolling contact with this common tan- 
gent, and hence with each otlier ; and their perimetral and 
ano'ular velocities will be the same as before. 




jFig. 18 



In Fig. 18, let OA, OB, be two axes which intersect at ; 
and let the two right cones OTD, OTF, be constructed on 
these axes, the cones having a common element OT. If 
through any point 3f in OT wq pass planes perpendicular to 
the axes OA and OB, the sections of the cones will be cir- 
cles which will be in contact at J/; and a constant velocity 
ratio will be maintained between the axes by means of these 
circles. For the angular velocities of these circles are, as 
before, 

^ ^ EE = 41 

a Mil BT' 



a constant ratio ; therefore the two cones will rotate in true 
rolling contact, and their angular velocities will be inversely 
proportional to the perpendiculars from any point of the 
common element on the axes. The relations of angular ve- 



MOTION BY ROLLING CONTACT. 



31 



locities, periods, etc., will evideiill}^ be the same as for two 
cylinders whose radii are in the same proportion as the radii 
of the bases of the cones. 

38. Having given the positions of the axes, and the ve- 
locity ratio, it is required to construct the cones. 




Fig. 19 

In Fig. 19, let OA be the driving axis, and OB the follow- 
ing axis : and let the velocity ratio of driver to follower be 

— = — ; in other words, OA is to make n revolutions while 
a 71 

OB makes m. revolutions. On OA lay off OC equal to n 
divisions on any convenient scale. Through C draw CD 
parallel to OB, and make it equal in length to m divisions of 
the same scale. Through D draw ODT, which will be the 
line of contact. From any point T of ODT, let fall the per- 
pendiculars AT and BT on the axes. If we now construct 
two right cones on these axes, having AT and BT as radii 
of their respective bases, these cones will roll together with 
the required velocity ratio ; for, from the figure, we have 



a; ^ m ^ sin COD ^ sin COD ^ AT , BT ^ AT 
a n sin ODC~sm BOD OT ' OT BT 



yyl ELEMENTARY MECPIANISxM. 

In other words, the radii of the bases have the required rela- 
tion. 

39. It is usual in practice to employ, not the whole cones, 
but only thin frusta of them, as shown in Figs. 19 to 24. 




ITig. SO 



In Fig. 19, the common element is located in the acute 
angle between the intersecting axes ; but it may as readily be 
placed in the obtuse angle, the location depending on the 




exact data of the problem. Examples of different arrange- 
ments are shown in Figs. 20, 21, and 22. In these figures the 
angles of intersection of the axes are the same as in Fig. 19, 



MOTION BY ROLLING CONTACT. 



33 



but the velocity ratio and tlie directional relation may be 
varied at pleasure. In Figs. 19 and 20 the velocity ratio is 
different, and the direction of rotation of the follower is also 




Fig 



changed in the latter by moving the element of contact from 
the acute to the obtuse anoie. In Fios. 21 and 22 the direc- 




Fig 



tional relation is the same as that of Fig. 20 ; but, by altering 
the velocity ratio, one of the cones becomes a flat disc in one 
case, and a concave conical surface in the other. 



34 ELEME^ITAllY MECHANISM. 

40. Thus far we have considered only those cases in which 
the axes intersect obliquely ; but in practice the axes intersect 
most frequently at right angles, as in Fig. 23. In this case 
it will be noticed that the cones are in contact along two ele- 
ments, OM and ON^ and that the followers will rotate in 
opposite directions. Thus, in Fig. 23, where A is the driver, 
the two followers, B and C, rotate in opposite directions, as 
shown by the arrows. But if, as in Fig. 24, the driving axis 
A be continued beyond the common vertex of the cones, and 
two other frusta be constructed, motion will be given to 
the two followers B and C in the same direction ; the velocity 
ratio of both pairs of frusta being the same. 

Axes neither Parallel nor Intersecting. 

41. Hyperboloid of Revolution. — When the axes do 
not lie in the same plane, motion may be transmitted from 
the one to the other by means of surfaces, known as hyperbo- 
loids of revolution. The hyperboloid of revolution is the 
warped surface generated by a right line revolving about 
another right line not in the same plane with the first. Its 
form and the manner of constructing it are shown in Fig. 25, 
both vertical and horizontal projections being employed for 
the sake of clearness. Let the axis be taken vertical ; it will 
be horizontally projected at 0' and vertically at (7c. The 
revolving line, or generatrix^ is, for convenience, taken in a 
position parallel to the vertical plane of projection, and is 
shown at MN^ M'N'. As this line revolves about the axis, 
any point, P, P , of the line describes a circle, whose radius 
is projected vertically at OP^ and horizontally in its true 
length at 0' P' . Draw the common perpendicular to the two 
lines MN and Cc. It will be projected horizontally in its 
true length at 0'D% and vertically in the point D. The circles 
described by different points of the line MN will evidently 
vary in size ; the largest being described by the points M and 



MOTION BY ROLLING CONTACT. 



35 



N respectively, and the smallest by the point D. To con- 
struct the projections of the curved surface, we must find the 



N H 




ITig. 35 



projections of the circle described by any point P of the line 
MK Its horizontal projection will be the circle W'P'E' ; 
while its vertical projection will be the straight line WPH, 



36 ELEMENTARY MECHANISM. 

and R and W will be points of the meridian curve. By re- 
peating this process for a sufficient number of points of the 
line MN. the meridian curve may be drawn ; and it will be 
found to be a hyperbola.. The circle A'D'B^, described by the 
point D^ (which is the intersection of the generatrix with 
the common perpendicular O'D') , is called the cirde of the 
gorge; and the circles described by the points Jf' and N^ are 
called tlie circles of the lower and upper bases respectively. 

If we take the line w?i, parallel to the vertical plane of 
projection, intersecting JfTV at i), and making angle nmc = 
angle NMc, and revolve it about (7c, we will evidently generate 
the same surface as before ; for the paths of m and M coin- 
cide, as do also those of n and jSf, and the point D is com- 
mon to both lines : hence any two points, one on each line, 
equidistant from D, such as P and Q, will describe the same 
circle. 

Through any point of the surface, then, two rectilinear 
elements, or generatrices, may be drawn ; and their projec- 
tions on a plane perpendicular to the axis will be tangent to 
the projection of the gorge circle on that plane. 

42. Rolling' Hyperboloids. — If through any point of 
a surface two lines of the surface be drawn, the plane which 
contains the tangents to both these lines will be tangent to 
the surface at that point. Hence, if through any point of 
the curved surface of a hyperboloid we pass two intersecting 
generatrices, the plane containing these two elements will be 
tangent to the surface at that point. The normal to the sur- 
face at that point must, of course, be perpendicular to that 
tangent plane ; and, as the surface is one of revolution, it 
must intersect the axis. 

If a series of such normals be drawn through different 
points of the revolving line, they will lie in planes perpen- 
dicular to the latter, and therefore parallel to each other. 
Suppose three planes to be drawn parallel to both the axis 
Cc and the generatrix 3fJSf', one through the axis, another 



MOTION BY ROLLING CONTACT. 37 

through the generatrix, and the third at any convenient dis- 
tance. Conceive a number of points to be laid off at definite 
and equal intervals on the line MN. Now, in passing along 
MN from one point to the other, the normal, though alwaj^s 
remaining perpendicular to 3/^, will still turn about the 
latter, so that its other end will describe on the plane through 
the axis a straight line; viz., the axis itself. Now, as these 
three planes are parallel, and the normal moves so that its 
two ends trace straight lines on two of the planes, it is evi- 
dent that the prolongation of the normal will trace a straight 
line on the third plane. This straight line may be taken as a 
new axis ; and by revolving MN^ the generatrix of the first 
hyperboloid, about this new axis, a second hyperboloid will 
be generated : and these two surfaces will, by construction, 
have a common normal at every point of the element of con- 
tact MN^ and will be tangent to each other all along that 
element. If one; of these hyperboloids be now rotated about 
its axis, it will drive the other by a mixture of rolling and 
sliding contact ; the sliding taking place in the direction of 
the element of contact, and the rolling in a direction perpen- 
dicular to that element. 

43. Velocify Fatio of Rolling- Hyperlboloicls. — In 
Fig. 26 we have two hyperboloids in contact along the line 
MN^ and revolving about the axes Oo and Rr respectively. 
Let P be the point of contact of the gorge circles APL and 
BPK^ and let the inclined hyperboloid be the driver. Then 
at any instant, any point on the gorge circle of either hyper- 
boloid will be moving in the direction of the tangent to that 
circle ; and the vertical projection of the tangent must evi- 
dently be perpendicular to that of the axis of the hyper- 
boloid. Draw Pa perpendicular to i^r, and of proper 
length to represent the velocity V of the point P of the 
driver at any instant. Similarly, draw Pb to represent 
the velocity v of the point P of the follower at the same 
instant. 



38 



ELEMENTARY MECHANISM. 





Fig. 26 



MOTION BY ROLLING CONTACT. 89 

Let a = angular velocity of driver, and 
a = angular velocity of follower. 



a 


= 


V 
PA 


V 
P'D' 


' 


= 


V 

PB 


V 




- pr^,^ 


a 
a 


= 


fx 


P'D 
P'B' 



(1) 



Draw the common normal to both surfaces at the point N. 
As the line MN is parallel to the vertical plane of projection, 
its vertical projection will evidently be perpendicular to that 
of the normal. Hence, for the vertical projection of the 
normal, draw ONR^ perpendicular to MN at N. and H 
being points on the axes, we readily find its horizontal pro- 
jection CyN'R'. As 0, iV, and R are respectively the ver- 
tical projections of three points on a straight line whose 
horizontal projections are 0', N\ and R'^ we have the ratio 

N'R' ^ NR 
O'N' ON 

But NR = PJVtan^, and ON = PiVtan <^, 
hence 

N'R' tan 



O'N' tan </) 



(-^) 



As B^0\ P^N\ and DR^ are parallel straight lines, we 
have 

P'D N'R' tan 



P'B' O'N' tan c/>* 



(3) 



P^D 

Substituting in (1) the value of ^ ^ , just found, we have 

P B 

i = -^ X 52El. (4) 



40 ELEMENTARY MECHANISM. 

The actual transmission of motion between the hyperbo- 
loids takes place by pure rolling contact in a direction per- 
pendicular to the common element passing through the points 
of contact ; for the sliding contact which takes place along 
that element can, of course, transmit no motion. Drawing 
PS perpendicular to MN^ and noting that in rolling contact, 
each pair of points of the surfaces which are in contact at a 
given instant, must at that instant be moving in the same 
direction with the same velocity, we have 

Fcos ^ = v cos <^, (5) 

V cosO 



V cos <^* 



(6) 



Substituting in (4) the value of — , just found, we have 



cos tan sin 



a cos (j) tan (f) sin ^ 
But sin = and sin 6 = — ^, hence 

a' ^ NH 

a EN' 



(7) 



(8) 



Hence, projecting both axes and the common element on 
a plane parallel to all of them, as in Fig. 26, we find that 
the angular velocities of the hyperboloids are inversely pro- 
portional to the sines of the angles made by the projection of 
the common element with the projections of the respective 
axes ; that is, inversely proportional to the projections of the 
perpendiculars let fall on the axes from any point of the 
common element. 

The radii of the gorge circles are directly proportional to 
the tangents of these angles ; that is, to the projections of 
the lines drawn from any point of the common element per- 
pendicular to the latter, and terminating in the axes. 



MOTION BY ROLLING CONTACT. 



41 



44. While the components of V and v along the line of 
contact must evidently have a constant value for all points of 
the same hyperboloid, yet as the angular velocities of all 
points of the hyperboloid umst be the same, it follows that 
the values of V and v themselves must increase as the points 
considered are farther from the gorge circle, and hence that 
the 'p^'^ceMage of sliding must decrease at the same rate. 

45. Having given the positions of the axes, and the 
velocity ratio, it is required to construct the hyperboloids. 




:Fig. 37 



In Fig. 27, let Rr, R'r', and Oo^ 0\ be the projections of 
the driving and following axes respectively ; the vertical 
plane of projection being taken parallel to both axes. Let 



42 ELEMENTARY MECHANISM. 

the driver be required to make n revolutions while the follower 

makes m revolutions ; in other words, — = —. 

a n 

On PR lay off PF equal to n divisions on any convenient 

scale. Through V draw VN parallel to Oo, and equal to 

m divisions of the same scale. From N\et fall ^A^'and Nil 

perpendicular to Oo and Mr respectively. Through JV and P 

draw the line NP. This will Ije the vertical projection of the 

element of contact ; for, from the triangles in the figure, 



m 


NV 


sin NPV 


sin NPV 


mi 


1 

a 


n 


PV 


sin PNV 


sin EPJSr 


EN 


a 



Through N draw ONR perpendicular to NP. ONR is the 
vertical projection of the normal ; hence the horizontal pro- 
jection of the point R must be at R^ on the horizontal 
projection, R'r\ of the axis Rr. Joining O'R' , we have the 
horizontal projection of the normal. Projecting N hori- 
zontally at N' on 0'R\ and drawing N'P' parallel to R'r\ 
we have the horizontal projections of the common element 
and the gorge circle radii O'P' and P'Q. We have thus 
determined all the data necessary to the construction of the 
hyperboloids, as explained in Art. 41. 

As in the case of cones, only thin frusta of these hyper- 
boloids (Fig. 26) are used in practice ; and their location is 
optional, except that, as already indicated, the percentage 
of sliding increases as they come nearer the gorge planes. 

46. Analogy between Cones and Hyperboloids. — • 
As the radii of the gorge circles are made smaller, the 
meridian curves of the hyperboloids will become flatter, and 
the surfaces will begin to approximate to the conical shape. 
AVHien the radii of both gorge circles reduce to zero, the 
axes will intersect, and the hyperboloids will become true 
cones ; the element of contact lying in the plane of the axes, 
and passing through their point of intersection. Cones, 



MOTION BY ROLLTXO CONTACT. 43 

then, may be considered as the limiting case of hj^per- 
boloids ; and it will be found, that, luider similar conditions, 
they will present similar peculiarities of arrangement. From 
the similarity of the solutions in Arts. 38 and 45, it is 
obvious that we may use our discretion in locating the 
common element in the case of the hyperboloids, just as 
explained in Art. 39 for the case of cones. By changing 
the common element from the acute angle between the 
projections of the axes, to the obtuse angle (a change similar 
to that shown by Figs. 19 and 20), we will change the 
directional relation of the hyperboloids. Again : by varying 
the velocity ratio so as to divide the angle in the same ratio 
as in Figs. 21 and 22, we will reduce one hyperboloid to a 
Jiat disc in one case, and to n holloiu hyperbolic surface in 
the other. 

47. The case of axes neither parallel nor intersecting 
may also be solved by means of two pairs of cones. 




Fis. 28 



In Fig. 28, let Aa^ Bh, be the driving and following axes 
respectively. Draw the line Cc intersecting the two axes in 
the points C and c, and let an intermediate axis be taken 
in this line. Now, a pair of rolling cones, d and e, having 
their common apex at C, will communicate motion from the 
axis Aa to the intermediate axis Cc ; and a pair of rolling 



44 ELEMENTARY MECHANISM. 

cones, /and r/, liaving their common apex at c, will transmit 
motion from the intermediate axis Cc to the axis Bh. By 
this means the rotation of Aa is transmitted, by pure rolling 
contact, to Bh. 

Let a, a^ and a' be the angular velocities of the axes ^a, 
Bh^ and Cc respectivelj, and i^, r, and li' the radii of the 
bases of their cones, those of the cones e and / being th.e 
same. Then 



a" R . a' R' , a! R 

a li a r a r 



exactly as if the cones d and g were in immediate contact. 



Practical Applications. 

48. We have now determined the theoretical forms re- 
quired to transmit motion by rolling contact with a constant 
velocity ratio, but the successful application of these forms 
in practice requires certain changes or substitutions to be 
made. It is impossible to transmit motion against any 
considerable resistance by means of such smooth surfaces, 
and hence various expedients are resorted to in order to 
obtain the necessary adhesion. 

49. Friction Gearing". — For light machinery, and in 
cases where a constant velocity ratio is not imperative, the 
rolling pieces may be made of different materials ; for 
instance, one may be made of wood and the other of iron. 
In this case, the iron wheel should be the follower. Again : 
one of the wheels may be covered with leather, or rubber, 
or other elastic material. To secure the necessary amount 
of adhesion in such cases, the rotating pieces are kept in 
contact and pressed together by adjusting their bearings, or 
applying weights or springs. 



MOTION BY ROLLING CONTACT. 



45 



50. Grooved Friction Gearing-. — Another method is 
shown in Fig. 29. The wheels are provided with angular 
grooves, shown in an enlarged section on the left. The 
angle between ab and cd is usually about forty to fifty 
degrees. The adhesion is greatly increased by this means, 
and is obtained, as before, by pressing the wheels together. 
Such wheels are widely used for hoisting-engines, and are 
generally made of cast-iron. 




Fig. S9 




51. Gearing. — The method in most general use for the 
prevention of slipping between rotating pieces is, to form 
teeth upon them. 

Gearing is the general term which includes all forms of 
mechanistic devices in which the motion is transmitted by 
means of teeth. The contact surface of the rotating pieces 
is called the pitch surface^ and its intersection with a plane 
perpendicular to the axis of rotation is termed the p^YcA line. 
This line is the basis of all calculations for velocity ratios 
and for the construction of teeth. The pitch line in the 
cases in which the velocity ratio is constant evidently 
becomes a pitch circle. 

52. Classification of Gearing". — Gearing is divided 
into classes according to the form of the pitch surfaces for 



46 



ELEMENTARY MECHANISM. 



which the toothed wheels are the equivalents. There are five 
such classes ; namely, spur gearing, bevel gearing, slcew gear- 
ing, screic gearing, and Jace gearing. 




Fig. 30 

53. In Spur Gearing-, illustrated by Fig. 30, the pitch 
surfaces are cylinders, and the teeth engage along straight 




ITiQ. 31 

lines which are parallel to the elements of the cylinders. A 
spur wheel having a small number of teeth is usually called 



Motion by rolling contact. 



4? 



a pinion. AYhen tlie teeth are formed on the mside of a 
ring, as shown in Fig. 31, the wheel is termed an annular 
luJieel. In this case, as before pointed out, the directions of 
rotation of driver and follower are the same ; while in the 
case of two spur wheels, the directions are opposite to each 




other. As the diameter of the pitch circle of a wheel 
increases, its curvature becomes less and less, and finally 
disappears when the former becomes infinite. In this case 
the toothed piece is called a rack (Fig. 32) , and its pitch line 
is the straight line tangent to the pitch circle of the wheel 
with which it works. In Figs. 30, 31, and 32, the various 
pitch lines are shown dotted. 




-Fi& 33 



54. In Bevel Gearing-, illustrated by Fig. 33, the 
pitch surfaces are cones, and the teeth engage along straight 



48 ELEMENTARY MECHANISM. 

lines the directions of which must all pass through the com- 
mon vertex of the two cones. In actual wheels, the teeth 
are, of course, placed all around the frusta ; but in the figure 
they are drawn only on part of the wheels, in order to 
show more clearly the relation in which they stand to the 
pitch surfaces. When the axes are at right angles, and 
two bevel wheels are constructed on equal cones, the line 
of contact making an angle of forty-five degrees with each 
axis, or, in other words, the velocity ratio being unity, the 
wheels are termed mitre gears. 

55. In Skew Gearing-, illustrated by Fig. 34, the 
pitch surfaces are hyperboloids of revolution. The teeth of 
these wheels engage in lines which approximate, in their 




general direction, to that of the common element of the two 
hyperboloids. This class of gearing is not often used, 
owing to the difficulty of forming the teeth ; the usual 
method for axes neither parallel nor intersecting being, to 
employ the intermediate cones described in Art. 47. 



Motion by rolling contact. 



49 



56. In Screw Gearing-, illustrated l)y Fig. 35, the 
pitch surfaces are cylinders whose axes are neither parallel 
nor intersecting ; and hence the cylinders touch each other at 




Fig. 35 

one x>oint only. The lines upon which the teeth are con- 
structed are helices on the surfaces of these cylinders. 
Motion is transmitted by a purely helical or screw-like 
motion. 

57. In Face Gearing, illustrated by Fig. 36, the teeth 
are pins usually arranged in a circle, and secured to a flat 



Q 



^u u u u y 



c 



imiE 



T?ig. 36 



circular disc fixed on the axis. Thus the contact is only 
between points of the surfaces of the pins. In Fig. 36 



50 



Elementary mechaMisM. 



the wheels are in planes perpendicular to each other, and 
the perpendicular distance between the axes is equal to the 
diameter of the pins, which in this case are cylindrical. 
This class of gearing is best adapted to wooden mill ma- 
chinery, and has been used for that purpose almost exclu- 
sively. 

58. Twisted Gearing'. — In Fig. 38 is illustrated 
another form of gearing, sometimes called tivisted gearing. 
It may be regarded as obtained from the stej^ped wheel 
shown in Fig. 37. The latter may be produced by cutting 
an ordinary spur wheel by several planes perpendicular to 




P^ig. 3V 



the axis, turning each portion through a small angle, and 
then securing them all together. By placing this wheel in 
gear with another, made in a similar manner, we combine 
the advantage of streno;th of laro-e teeth with the smooth- 
ness of action of small ones. If the number of cutting 
planes be indefinitely increased, and each section be turned 
through an exceedingly small angle, it is clear that a twisted 
wheel, such as shown in Fig. 38, will be the result. But 
instead of ordinary spur teeth, whose elements are parallel 
to the axis of the wheel, we now have teeth whose elements 
have the directions of helices. The result is, that, in addition 



MOTIOiSf BY KOLLIXG CONTACT. 



51 



to the pressure prodiiciDg the rotation, there will be a 
pressure produced in the direction of the axis, tending to 



slide the wheels out of gear. 





E^ig. 39 



The endlong pressure on the bearings may be prevented 
by the use of a wheel such as is shown in Fig. 39. By this 
arrangement, there is no longitudinal pressure on the bear- 
ings whatever, and the wheels run in gear with a smoothness 
of action unsurpassed by any other kind of gearing. 



5^ ELEMENTARY MECHANISM. 



CHAPTER lY. 

COMMUNICATION OP MOTION BY ROLLING CONTACT. 

VELOCITY RATIO VARYING. 

DIRECTIONAL RELATION CONSTANT. 

Logarithmic Spirals. — Ellipses. — Lobed Wheels. — Intermittent 
Motion. — Mangle Wheels. 

59. It has been shown (Art. 32) that, in the rolling con- 
tact of curves revolving in the same plane about fixed parallel 
axes, the point of contact always lies in the line of centres. 
The radii of contact coincide with this line ; and at the point 
of contact the curves have a common tangent which must 
make equal angles, on opposite sides of the line of centres, 
with the two radii of contact. 

60. In the preceding chapter, the ratio of the radii of 
contact was constant, and hence the velocity ratio was con- 
stant. If the curves are such that the radii of contact vary, 
the point of contact moving along the line of centres, the 
velocity ratio must vary. The sum of the lengths of each 
pair of the radii of contact must evidently be constant if the 
point of contact lies between the axes, or their difference 
must be constant if the axes lie on the same side of the 
point of contact. 

*61. The Log-aritlimic Spiral is a curve having the 
property, that the tangent makes a constant angle with the 
radius vector. Let two equal logarithmic spirals be placed 
in reverse positions, and turned about their respective poles 



MOTION BY ROLLING CONTACT 



m 



hs fixed centres until the curves are in contact. Each of the 
radii of contact is a radius vector of the curve in which it 
lies, and hence both radii make the same angle with the 
common tangent at the point of contact. But this can only 
be true if the radii of contact lie in one straight line, namely, 
the line of centres ; in other words, the point of contact lies 
on the line of centres, and equal logarithmic spirals are 
therefore rolling curves. 

*62. To Construct the Log^arithmic Spiral. — In Fig. 
40, let be the pole of the spiral, and let A and B be two 




Fig. 40 



points through which it is desired to draw the curve. From 
the property of the curve given above, namely, that the tan- 
gent makes a constant angle with the radius vector, it may 
readily be proved that, if a radius vector be drawn bisecting 
the angle between two other radii vectors, the former will be 
a mean proportional between the two latter. Draw the radii 
vectors AO and BO, and the line QD bisecting the angle 
AOB. Then, if Z) is a point of the curve, OD must be a 
mean proportional between OA and OB ; in other words, 

QA^QR, On the straight line AO lay off OC = OB. On 

AOC as a diameter, describe the semi-circle AEC. Draw 
OP perpendicular to AOC. Then OE is a mean proportional 



54 



ELEMENTARY MECHANISM. 



between OA and OB. Therefore make OD = OE, and D 
will then be a point on the curve. 

In the same manner, bisect the angle AOD^ make OF a 
mean proportional between OA and OD to find the point F, 
and so on. 

63. Since the logaritlimic spiral is not a closed curve, two 
such spirals cannot be used for the transmission of continu- 
ous rotation ; but they are well adapted for reciprocating 
circular motion. 

In Fig. 41, let the distance between the axes A and B be 
given ; and let it be required, that, while the driving axis A 
turns through a given angle, the velocity ratio shall vary 
between given limits. 




Fig. 43-. 



Divide AB at T into two segments whose ratio is one of 
the given limits, and at C into segments whose ratio is the 
other limit. Lay off the angle DAC equal to the, given angle, 
and make AD = AC. 

The problem is now simply to construct a logarithmic spiral 
(Art. 62) having the pole yl, and passing through the points 
T and D. 

The follower is necessarily a portion of the same curve in 
a reverse position ; and the latter having been drawn about 



MOTION BY ROLLING CONTACT. 



55 



the pole jB, draw arcs of circles about B with the radii BC 
and BT. The portion of the curve between the intersections 
of these arcs and the spiral will be the required edge of the 
follower. 

Let a = angular velocity of driver, and a = angular 
velocity of follower ; then, while the driver turns from the 
position in the figure through the angle TAD, the velocity 

AC 
BC 



/I T' 

ratio will vary between the limits — = and — 

-^ a BT a 



64. Rolling Ellipses. — In Fig. 42, let ETH and FTG 

be two similar and equal ellipses, placed in contact at a point 




E^ig.-^^S 



T, such that the arcs ET and FT are equal , E and F being 
the extremities of the respective major axes. It is a prop- 
erty of the ellipse that the tangent CTD makes equal angles 
with the radii BT and bT, or AT and aT Therefore the 
angle DTA = angle CTB, and angle BTh = angle CTa-, 
hence BTA and hTa are straight lines. Also, since the arc 
ET = arc FT by construction, TA and Tb are equal ; 
therefore BT + TA = BT + Tb = FG = EH, a constant 
length whatever the position of the point of contact, T. 
Similarly, bT -\- Ta = FG = EH. Hence two equal and 



56 ELEMENTARY MECHANISM. 

similar ellipses can transmit motion between parallel axes 
b}^ pure rolling contact ; each ellipse turning about a focus as 
a fixed centre, and its major axis being equal to the dis- 
tance between those centres. The velocity ratio will in this 

case vary between the limits — = = and — = 

•^ a BG AE a 

AJT' AV 

— — = — — , the two limits being reciprocals of each other. 
FB A.H 

65. LiObed Wheels. — By using rolling ellipses, as 
shown in the preceding article, we can obtain a varying ve- 
locity ratio having one maximum and one minimum value, 
during each revolution. 

But it may be necessary that there shall be two, three, or 
more maximum, alternating with as many minimum, values 
of the velocity ratio during each revolution. 

Lohed icheels which will roll together and answer these 
conditions can be produced by several methods from the 
logarithmic spiral and the ellipse. 

66. Lobed Wheels derived from the Log^arithiiiic 
Spiral. — In Fig. 43, let A and B be two fixed parallel axes, 
and let it be required to communicate motion between them 
by wheels so constructed that the velocity ratio will have 

/ 7? T' 

four maximum and four minimum values. Let — = be 

a AT 

one limit : then the other is necessarily the reciprocal of this, 

a' AT 
""' -a'^BT' 

Make the angles TAC and DBT equal to 45°. Make 
BD = AT and AC = BT. Construct (Art. 65) the por- 
tion CT of a logarithmic spiral having A as the pole, and 
passing through the points G and T. Draw CF, TE^ and 
TZ>, similar curves symmetrically placed with regard to BT 
and AC, We have thus constructed one lobe of each wheel ; 
and, as the angles TAF and DBE each include one-fourth 
of a circumference, the quadrilobes can be completed as 



MOTION EY POLLING CONTACT. 



57 



shown, and will roll together with the varying velocity ratio 
required. 




P"ig. 43 



The angles TAF and DBE may include any aliquot part 
of a circle ; hence pairs of wheels with any desired number 
of lobes can be made in this way. They will roll together in 
similar pairs, unilobe with unilobe, bilobe with bilobe, and 
so on ; but dissimilar pairs, such as one bilobe and one 
trilobe, will not roll together. 

67. Lobed Wheels derived from the Ellipse. — 
Lobed wheels may be derived from rolling ellipses by the 
method of contracting angles, as illustrated by Fig. 44. 

Let A and B be the fixed foci of two equal rolling ellipses 
in contact at T. Draw the radii A\^ A'2, etc., dividing the 
semi-ellipse T6 into equal angles about the focus A, and con- 
sequently into unequal arcs. If we describe arcs about T 



58 



ELEMENTARY MECHANISM. 



through the points 1, 2, 3, etc., cutting the other semi-ellipse 
T^' at the points 1^ 2^ 3', etc., it is evident that the arc 
T\ = TV, T2 = T2\ TS = T3', etc. Therefore the points 
1 and 1', 2 and 2', etc., will come in contact on the line of 
centres AB ; and AB = Al -\- BY = A2 + Br =, etc. Bi- 
sect the angle TAX by the line AI^ and bisect the angle TBV 
by the line BI'. Make AI = Al, BI' = BV. It is evident 
that these points, / and /', will come in contact on tlie line of 
centres when they have turned through the angles TAI 
(=1 angle 2M1) and TjBr(= i angle TBV) respectively. 




Thus, if we find the series of points /, J/, III, etc., and 7', 
//', Iir, etc., in the manner just described, and draw through 
them two curves, as shown in the figure, they will be quad- 
rants of two similar and equal bilobes, of which the remain- 
ing similar portions can then be readily drawn. From the 
above considerations, it is evident that these bilobes will roll 
together in perfect rolling contact. The velocity ratio will 

vary between — = and — = — — . By contracting the 



BT a A'. 

angles to one-third, we can form the outlines of a pair of 
trilobes, and so on. 



MOTION BY ROLLING CONTACT. 



59 



The wheels thus outlined will roll together iu similar pairs, 
as bilobe with bilobe, trilobe with trilobe, and so on ; but 
dissimilar pairs, such as one bilobe and one trilobe, will not 
roll together. 

*68. Interchangeable Lobecl Wheels. — In Figs. 45 
and 46 is illustrated a method of constructing lobed wheels 
from an ellipse, by which any tico wheels of the set will roll 
together. The process of construction is simple and practi- 
cal ; but the rolling properties of the curves do not admit of 
simple demonstration, although they may readily be proved 
by graphical construction. In Fig. 45, let A and B be the 



n M L K 



ng. 45 




foci of an ellipse, CGPV. Describe a circle about its centre 
with a radius equal to the semi- focal distance OA. Draw 
the indefinite tangent UN parallel to BA. With radius 0(7, 
equal to the semi-major axis, and centre 0, describe an arc 
CA", and lay off on the tangent the lengths KL, LM, and 3IJSf, 
equal to UK. From the centre lay off on OF the dis- 
tances 00 = OK, OB = OL, OE = OM, and so on. With 
OC, OB, OE, etc., as semi-major axes, describe a series of 
concentric ellipses, having the common foci A and B. The 
primary ellipse is the curve required for the unilobe ; the 
second ellipse, BQ, is the basis for the bilobe ; the third, 
ER, for the tiilobe ; the fourth, FS, for the quadrilobe ; and 



60 



ELEMENTARY MECHANISM. 



SO on. Draw a semi-circle about A, and divide it into any 
number of equal angles by equidistant radii. 

To form the bilobe (Fig. 46), divide a quadrant into the 
same number of equal angles as tlie semi-circle is divided, 
and on the equidistant radii in the quadrant lay off BV = A\ , 
52' = ^2, etc. Through the points 1', 2', 3^ etc., draw a 
curve : this will be one-fourth of the bilobe ; the remaining 
portion of which, being symmetrical, can readily be drawn. 




For a triloba, an angle of 60° is similarly divided, and the 
proper distances laid off on the equidistant radii in that angle. 
For a quadrilobe, we use an angle of 45°, and so on. 

The velocity ratio of any two of these wheels in gear will 
vary between two limits, one of which will be the longest 
radius of the driver divided by the shortest radius of the 
follower, and the other the shortest radius of the driver 
divided by the longest radius of the follower. 

69. Compulsory Rotation of Kollinsr Ellipses. — In 
the case of rolling ellipses (Fig. 42) , it is evident that, when 
the motion takes place in the direction of the arrows, the 
radius of contact of the driver is increasing from AE to AH^ 
and hence motion can be readily transmitted from the axis A 



MOTION BY ROLLING CONTACT. 



61 



to the axis B. But, when H has passed G^ the radius of the 
driver is decreasing^ and the driver will therefore tend to 




Fig. 47 



leave the follower. This can be prevented by forming teeth 
on the rolling faces of both pieces ; but, if this is done, we 
no longer have pure rolling contact. 




I^ig. 48 



"When the position of the pieces will allow it, we can con- 
nect the free foci by means of a link, as in Fig. 47, since 



62 



ELEMENTARY MECHANISM 



(Art. 64) the distance between the free foci is constant in 
rolling ellipses. There will, however, be times during the 
revolution when the link will be in line with the fixed foci, 
and hence cannot transmit motion. This necessitates the 
formation of teeth on a small portion of each ellipse, near 
the ends of the major axis, as shown in Fig. 47. 

Another method, when the revolution always takes pla(ae 
in the same direction, is, to form teeth on the retreating edge 
of the driver and the corresponding edge of the follower. In 
this case it is necessary to provide some means of insuring 
the proper contact of the teeth in order to prevent jamming. 
This may be done, as shown in Fig. 48, by placing a pin on 
the driver and a guide plate on the follower, which arrange- 
ment compels the first tooth to enter the proper space. 

70. Intermittent Motion. — It may happen that the 
variation in the velocity ratio is to consist of an intermittent 




OFig. 49 



motion of the follower, while the driver revolves uniformly. 
In Fig. 49 is shown an intermittent motion formed from 
two spur wheels by cutting away the teeth of the driver on a 



MOTION BY ROLLING CONTACT. bS 

portion of the circumference. There is the same objection 
to this method as before mentioned for elliptical wheels ; 
namely, that the teeth are apt to jam after a period of rest 
of the follower. A partial remedy is the application of a pin 
and guide plate, similar to the arrangement shown in Fig. 48. 
A more complete motion is shown in Fig. 50. A portion of 




the driver is a plain disc of a radius greater than the pitch 
circle of the driver. A portion of the follower is cut away, 
to correspond to this ; so that, while there is a slight clear- 
ance between the two faces, the follower is prevented from 
turning until the pin and curved piece come in contact. 

Velocity Ratio Varying. Directional Relation Changing. 

71. Mangle Wheels. — By combining a s^nir wheel with 
an annular wheel, we obtain a mangle ivJieel, as shown in 
Fig. 51. The direction of rotation is changed by causing the 
pinion, which always revolves uniformly in the same direc- 
tion, to act alternately on the spur and on the annular portion. 



64 ELEMEI^TARY MECHANISM. 

The velocity ratio is constant during each partial revolution 
of the mangle wheel ; but it is changed each time that the 
pinion passes from the spur to the annular portion, and vice 




I^ig. 51 

versa. The pinion is mounted so that its shaft has a vibra- 
tory motion, working in a straight slot cut in the upright bar. 
The end of the pinion shaft is guided in the groove CD^ the 




centre line of which is at a distance from the pitch lines of 
the mangle wheel equal to the pitch radius of the pinion. 
The pinion may also be mounted in a swinging frame, as 
indicated by dotted lines. 



MOTION BY ROLLING CONTACT. 



65 



If we construct the teeth of the spur and annular portions 
of the mangle wheel on the same pitch line, as in Fig. 52, we 
will obtain a combination in which the velocity ratio is con- 
stant; the directional relation changing, as in the preceding 
arrangement. 

72. Mangle Rack. — A rack can be made in a similar 
manner to the above, and a reciprocating motion obtained 
from continuous rotation. Such motion is, however, more 







simply obtained by means of the pinion and double rack, 
shown in Fig. 53. Pins are placed on a portion of the face 
of the pinion, which engage with the pins of the rack above 
and below alternately, driving the rack back and forth. 



66 ELEMENTARY MECHANISM. 



CHAPTER V. 

COMMUNICATION OF MOTION BY SLIDING CONTACT. 

VELOCITY RATIO CONSTANT. 

DIRECTIONAL RELATION CONSTANT. 

TEETH OF WHEELS. 

Special Curves. — Rectification of Circular Arcs. — Construction oj 
Special Curves. — Circular Pitch. — Diametral Pitch. 

73. General Problem It has been shown (Art. 32) 

ihat, in order to obtain a constant velocity ratio in contact 
motions, the axes of the pieces being parallel, the curves 
must be such that their common normal at the point of 
contact shall always cut the line of centres at the same point. 
The curved edge of one of the moving pieces may always 
be assumed at pleasure ; the problem then being to find such 
a curve for the edge of the other, that, when motion is 
transmitted by the contact of these curved edges, the velocity 
ratio of the two axes may be constant. This problem is 
always capable of solution, theoretically at least; and, as 
the assumed curve may be of any shape whatever, we can 
obtain an infinite number of pairs of such curves. For 
practical purposes, there are certain definite curves in almost 
universal use, and these will be first discussed. In Chap. III. 
has been explained the method of finding the diameters of 
two pitch circles, which by their rolling contact shall trans- 
mit motion with a given velocity ratio. We now propose to 
show how to describe certain curves, which, when substi- 



MOTION BY SLIDING CONTACT. 



67 



tilted for the circles, and caused to move each other by 
sliding contact, shall exactly replace the rolling action of 
the circles, so far as relates to the production of a constant 
velocity ratio. 

74. Epicycloid and Hypocycloid. — In Fig. 54, let 
A and B be the centres of motion of the drh^er rnd follower 

respectively, and let — be the required velocity ratio. 



Divide the line of centres at T, so that 



AT 
BT 



^. Tl;«n, if 
a 



with radii AT and BT we describe two pitch circles, 3/JV 
and BS, as shown, these two circles ^dll roll in coatact 
with the required velocity ratio. 




TT-ig. 54= 



Let a describing circle be taken of any radius, such as 
cT, and with it describe an epicycloid Td by rolling it on 
the outside of the pitch circle MJSF, and a hypocycloid Te by 
rolling it on the inside of the pitch circle BjS. If these 
curves be used for the curved edges of two pieces whose 
centres of motion are A and B respectively, and the lower 
one be rotated to the position aa/, it will drive the other so 



QS ELEMENTARY MECHANISM. 

as to bring it to tlie position bb' ; for, by the known 
properties of the curves, the}^ will have their point of con- 
tact, P, in the circumference of the describing circle when 
its centre c is on the line of centres, AB, and they will also 
have a common normal and a common tangent at that 
point. Draw the line TP from the point of contact of the 
two pitch circles to the point of contact of the two curves. 
Now, on whichever of the two pitch circles we regard the 
describing circle to be rolling at the instant, its instan- 
taneous centre of motion will evidently be the point T. 
For that instant, then, the point P revolves about T; that 
is, it moves in a direction perpendicular to TP, and hence 
the line TP is the common normal to the two curves at 
that instant. Of course, this same argument may be applied 
to any other position of the curves in contact ; and, as their 
normal thus always cuts the line of centres in the fixed 
point T, it is evident that these curves will transmit motion 
with a constcmt velocity ratio. Furthermore, as the arc Ta 
= arc TP, and as the arc Tb = arc TP, we have arc Ta = 
arc Tb ; showing that the velocity ratio will be the same as 
that of the two pitch circles. By transmitting motion by 
sliding contact, then, between these two curves, we may 
exactly replace the rolling action of the two pitch circles, 
as far as the velocity ratio is concerned. 

75. Epicycloid and Radial Line. — In Fig. 54 the 
diameter of the describing circle is less than the radius BT. 
But this is not a necessary condition. If we change its 
diameter, we will change the shape of both curves ; but the 
two curves generated by the same describing circle will 
always work together. 

If we take the diameter of the descril^ing circle just equal 
to the radius BT, we will get a special case of the hypo- 
cycloid. Under these conditions (Fig. 55) the latter will 
become a straight line passing through the centre B. All 
the arguments of the last article apply to this case as well ; 



MOTION BY SLIDING CONTACT. 



69 



aud we thus see that in this case an epicycloidal curve 
turning about A, and a radial piece turning about B, will, 
by sliding contact, transmit motion with the same velocity 
ratio as the pitch circles. 




76. Epicycloid and Pin. — In Fig. 54 the convexity 
of the two curves lies in the same direction, and they lie 
on the same side of the common tangent. In Fig. 55 
the hypocycloid has become a straight line coinciding with the 
tangent of the epicycloid. If we increase the diameter of 
the describing circle still more, the two curves will have 
their convexities in opposite directions, aud they will lie on 
opposite sides of the common tangent. As the describing 
circle becomes larger and larger, the hypocycloid becomes 
more and more convex, and decreases in size, until, when 
the describing circle is taken with the same diameter as the 
pitch circle BS, the hypocycloid will degenerate into a mere 
point, the tracing point itself. If, then (Fig. 56), we 
assume a pin to be placed at the point P in the circum- 
ference of as (the diameter of the pin being so small that 
the latter may be considered as a mere mathematical line) , 



70 



ELEMENTARY MECHANISM. 



it follows that it inaj be driven by the epicycloid Pa with 
the same constant velocity ratio as the pitch circles. 




Fig. S6 



77. Involutes. — In Fi^. 57, let A and B be the centres 



of motion, make 



AT 



— , and describe the pitch circles 



BT a 

MN and RS^ as before. Through T draw the straight 
line DTE inclined at any angle to the line of centres ; from 
A and B drop the perpendiculars AD and BE upon DTE. 
With these perpendiculars as radii, and A and B as centres, 
describe the circles M'N' and R' S\ which will evidently be 
tangent to the line DTE. Through the point T describe 
the involute aTd on the base circle M'l^\ and the involute 
IjTq on the base circle B!S' . If these curves be used for 
the edges of two pieces whose centres of motion are A and 
B respectively, and the lower one be rotated to the position 
a'Pd\ it will drive the other to the position h'Pe\ For 
any line tangent to either base circle will evidently be 
normal to the involute of that circle. Now, when the curves 
are in contact, the normal to the involute of M'N' must be 



MOTION BY SLIDIXO CONTACT. 



n 



a line drawn from the point of contact tangent to M'N' ^ 
and the normal to the involute of R'S' must be a line drawn 
from the point of contact tangent to R'S^. But, as the 
curves must be tangent to each other at the point of contact, 
they must have a common normal at that point. This 
common normal must evidently be tangent to hoth base 
circles, and must hence be the line DTE. The point of 
contact, then, alwa3^s lies in the straight line DTE \ and 
as the latter is the common normal, and cuts the line of 
centres in the fixed point T, the velocity ratio is constant, 
and is equal to that of the base circles. 




But, from similar 



,., BE BT 

trianojles, — — = — — : 
AD AT 



that 



the 



velocity ratio of the pitch circles is the same as that of the 
base circles. Hence the involutes, as described, will by 
sliding contact transmit motion with the same velocity ratio 
as the pitch circles would by rolling contact. 

78. General Solution. — The four methods just de- 
scribed are the ones most generally employed in the practical 
solution of the problem of securing a constant velocity 



72 



EL15MENTARY MECHA^-ISM. 



ratio in sliding contact motions. But we are not by any 
means limited to the curves above given. Instead of a 
describing circle^ we may use a describing curve of any shape, 
provided only that its radius of curvature never exceeds in 
length the radius of the circle in which the curve is to roll, 
and thus generate an infinite number of pairs of curves that 




will satisfy the given condition. Thus, in Fig. 58, let ^4, 5, 
and T be taken as before, and draw pitch circles MN and 
RS. Now, if we take any curve, such as HTP, and roll 
it on the outside of one pitch circle and on the inside of the 
other, any point of this describing curve will generate two 
curves which will give the desired velocity ratio by sliding 
contact. For, let the describing curve be in the position 
shown, being in contact with the pitch circles at T \ and let 
P be the describing point. The straight line TP will be the 
common normal to the two curves, because, on whichever 
of the two pitch circles we regard the describing curve to 
be rolling at the instant, the point of contact, T, is the in- 
stantaneous centre of motion ; so that the motion of P in 



Motion by sliding contact. TS 

either curve is perpendicular to TP. As the point of 
contact of the two curves is alwa^^s in the describing curve, 
the same argument is true for any point of contact. As the 
common normal will thus always pass through the same 
point, T, of the line of centres, AB^ these curves will, by 
moving in contact, produce the desired velocity ratio, 
exactly replacing the rolling action of the two pitch circles. 

79. Conjug-ate Curves. — Any two curves so related, 
that, by their sliding contact, motion will be transmitted with 
a constant velocity ratio, are called conjugate curves. Any 
curve being assumed at pleasure, we may proceed to find 
another curve, so that the two curves will be conjugate to 
each other. If, for instance, in Fig. 58, the curve Pa be 
given, it is only necessary to find the shape of the curve, 
HTP^ which, by rolling on the outside of MN^ will generate 
Pa. By then rolling this describing curve HTP on the 
inside of RS.^ we will obtain the required curve, Ph. Again : 
had Ph been given, we could, by a similar process, have 
found Pa ; and Pa and Ph are conjugate curves. 

The labor of finding the shape of this describing curve, 
and using it in this manner, is, however, generally very 
considerable ; so that, for practical purposes, the following 
simple and satisfactory mechanical expedient, due to Pro- 
fessor Willis, is usually resorted to. 

In Fig. 59, ^ and B are a pair of boards, whose edges 
are formed into arcs of the given pitch circles. Attach to 
A a thin piece of metal, (7, the edge of which is cut to the 
shape of the proposed curve a5, and to 5 a piece of draw- 
ing paper, D ; the curved piece being slightly raised above 
the surface of the board to allow the paper to pass under it. 
Roll the boards together, keeping their edges in contact, so 
that no slipping takes place ; and draw upon D, in a suffi- 
cient number of positions, the outline of the edge ah of C. 
A curve, de, which touches all the successive lines, will be 
the corresponding curve required for B. 



u 



ELEMENTARY MECPlAXlsM. 



For, l\y the vei-y mode in which it lias been obtained, it 
will touch ah in every position ; hence the contact of the 
two curves ab and cle will exactly replace the rolling action 
of the two pitch circles. To prevent the boards from 
slipping, a thin band of metal, such as a watch spring, may 
be placed betw^een them, being fastened to B at ^, and to 




±nig. SO 



A at h. The respective radii of the circular edges of the 
boards must, in that case, be made less than those of the 
given pitch circles by half the thickness of the metal band. 

80. The solutions given above may be used to find the 
curved edges of any two pieces transmitting motion by 
sliding contact with a constant velocity ratio, but by far 
their most important application is in finding the proper 
shapes for the teeth of wheels. 

We shall now give the methods of laying out on paper 
the principal curves employed for that purpose, and then 
proceed to examine their practical application in the forma- 
tion of teeth. 

81. Rectification of Circular Arcs. — In construct- 
ing these curves, as well as in many other graphic operations, 
it becomes necessary to determine the . lengths of given 



Motions by sliding contact 



75 



circular arcs, as yn'cII as to la}^ off circular arcs of given 
lengths. Either of these problems may, of course, be solved 
by calculation ; but for our purposes it is much more satis- 
factory to emplo}^ the following elegant and surprisingly 
accurate methods of approximation, devised by Professor 
Rankine. 

I. To rectify a given circular arc; that is, to lay off its 
length on a straight line. 




Fig. 60 

In Fig. 60, let AT he the given arc. Draw the straight 
line BT tangent to the arc at one extremity, T. Bisect the 
chord AT ixt D, and produce it to (7, so that TC = DT — 
AD. 

With C as a centre, and radius AC, describe the circular 
arc AB, cutting BT at B. Then BT is the length of the 
given arc AT, very nearly. 

II. To lay off, on a given circle, an arc equal in length to 
a give?! straight line. 




In Fig. 61, let r be the point desired for one extremity 
of the arc. Let BT, drawn tangent to the circle at T, be 
the given straight line. Lay oft CT = {BT. With O as a 
centre, and radius BC^ describe the circular arc BA, cutting 



76 tlLF.MENTAilY MECMANtSM. 

the given circle at A. Then the arc AT is equal in length 
to the given straight line BT, very nearly. 

It follows that, to lay off on a given circle an arc equal to 
a given arc on another circle, we must first rectify the given 
arc according to I., and then lay off according to II. the 
required arc equal to the length so found. 

82. Degree of Accuracy in Above Processes. — The 
error in each of these processes consists in the straight line 
being a little less than the arc. But this difference is very 
slight, amounting to only -g^o of the arc when the latter is 
60°. The error varies as the fourth power of the angle, so 
that it may be reduced to any desired limit by subdivision. 
Thus, for an arc of 30°, the error will be -^-^ x (fj)^ = 
T4 4T0"- S^ long, then, as we use these processes for arcs 
not exceeding 60°, the results will be abundantly accurate 
for all practical purposes. When the arcs exceed 60°, sub- 
division should be resorted to. 

83. Construction of the Epicycloid. — In Fig. 62, 
let C be the centre and CT the radius of a circle rolling on 
the outside of the Jixed circle whose centre is A and whose 
radius is AT. Any point in the circumference of the rolling 
circle will describe a curve, which is known as an epicycloid. 
Let it be required to draw the curve described by the point 
T of the rolling or describing circle. 

Divide the semi-circumference of the latter into any 
number of equal arcs, Tl', 1'2', 2^3', etc., and through the 
points of division, 1', 2', etc., and also through (7, describe 
arcs of circles about ^ as a centre. Lay off on the fixed 
circle (Art. 81) the arcs Tl = TV ; 1, 2 = 1', 2'; 2, 3 = 
2\ 3', etc. ; and through the points of division, 1, 2, 3, etc., 
draw radii from A^ and produce them. 

As the describing circle rolls along the fixed circle, its 
centre will successively occupy the positions Cj, Cg, Cg, etc. 
If we draw the describing circle with its centre in any one 
of these successive positions, as c^, its intersection b with 



MOTION BY SLIDING CONTACT. 



the circular arc through 2' will be a point of the epicycloid 
required. Similarly, we obtain the points a, f7, e, /, g \ 
and the curve drawn through these points will be the epi- 
cycloid required. If greater accuracy is required, we need 
only increase the number of arcs into which we have divided 
the describino; circle. 




S"ig. 63 



This method of finding points of the curve is objectionable 
on account of the resultant obliquity of the intersections at 
a and /. This may be avoided, and the construction sim- 
plified, by laying off the arc lb = 7r2', mrZ = ^'3', etc. In 
this case it is not necessary to construct the rolling circle 
in its various positions ; and, as this method gives the best 
results for points of the curve near T (which is the part of 



?§ 



ELEMENTARY MECHANISM. 



the curve emploj^ed in teeth of wheels), it is greatly to be 
preferred for practical work. 

84. Construction of the Hypocycloid. — The hypo- 
cycloid is the curve described by a point in the circumference 
of a circle rolling on the inside of a fixed circle. Its con- 
struction, shown in Fig. 63, is in every way similar to that 
of the epicycloid. 

5/ 




When the diameter of the rolling circle is less than the 
radius of the fixed circle, the curve lies on the same side of 
the centre A as the successive points of contact of the two 
circles. When the diameter of the rolling circle is greater 
than the radius of the fixed circle, the curve lies on the 
opposite side of the centre A, When the diameter of the 



MOTION BY SLIDING CONTACT. 



79 



rolling circle is equal to the radius of the fixed circle, as 
shown on the left in Fig. Q>o^ the radii ^12, J. 3, etc., pass 
through the points 2", 3^ etc., and the points h and /<, A: and 
fZ, etc., coincide so that the curve becomes a straight line ; 
and this line is a radius of the fixed circle. 

85. Construction of the Cycloid. — The cycloid is the 
special case of the epicycloid and h^^pocycloid, in which the 
radius of the fixed circle becomes infinite, and the circum- 
ference of the circle a straight line. The cycloid is thus 
described by a point in the circumference of a circle rolling 
on a straight line. Its construction is in all respects similar 
to that of the epicycloid and hypocycloid. 

86. Construction of the Involute. — The involute is 
generated 'by a point in a straight line which rolls along a 




fixed circle ; or we may regard it as formed by the end of a 
thread which is unwound from about the circle, and kept taut. 
It will thus always lie in the direction of a tangent to the 



80 ELEMENTARY MECHANISM. 

circle. Hence, to construct the curve, draw any number of 
tangents to the base cu'cle, and on them lay off the rectified 
arc of the circle from the point of tangency to the point on 
the circle where the involute begins. In F'ig. 64, then, make 
«2 = arc 1, 2 ; ?>3 = arc 1, 3, etc. The curve drawn through 
the points 1, a, 6, c, etc., will be the required involute. 

87. Circular Pitch Having divided the line of cen- 
tres, in any given case, according to the assigned velocity 
ratio, and described the pitch circles, we must next divide 
the circumference of each pitch circle into as many equal 
parts as its wheel is to have teeth. The length of the circu- 
lar arc measuring one of these divisions is called the circular 
pitcJi, and often simply the j^itch, of the teeth. Circular 2^ itch, 
then, is the distance, measured on the circumference of the 
pitch circle, occupied by a tooth and a space. This pitch 
must evidently be the same on both pitch circles. The num- 
bers of the subdivisions, and hence the numbers of teeth, 
are proportional to the diameters of the pitch circles ; and, a 
fractional tooth being impossible, the pitch must be an 
aliquot part of the circumference of the pitch circle. 
Let F = circular pitch of the teeth in inches ; 

I) = pitch diameter, i.e., diameter of pitch circle in 

inches ; 
iV = number of teeth ; 

TT = ratio of circumference of a circle to its diame- 
ter = 3.141G. 
Then 

NF ^ ttD, 
and hence 

P ' TT ' N 

From the above relatione, we may evidently find any one 
of the three elements P, i>, JSf; the other two having been 
given by the problem. 



MOTION BY SLIDING CONTACT. 



81 



For convenience in calculation, the following table is ap- 
pended, in which the pitch diameters are calculated for a 
pitch of one inch. 



PITCH DIAMETERS. 

FOR OXE INCH CIRCULAR PITCH. 



No. 

of 

Teeth. 


Pitch 


No. 

of 

Teeth. 


Pitch 


No. 

of 

Teeth. 


Pitch 


No. 

of 

Teeth. 


Pitch 


Diameter. 


Diameter. 


Diameter. 


Diameter. 





2.86 


32 


10.19 


55 


17.51 


■ 78 


24.83 


10 


3.18 


33 


10.50 


50 


17.83 


79 


25.15 


11 


3.50 


34 


10.82 


57 


18.14 


80 


25.46 


12 


3.82 


35 


11.14 


58 


18.40 


81 


25.78 


13 


4.14 


36 


11.40 


59 


18.78 


82 


26.10 


14 


4.46 


37 


11.78 


00 


19.10 


83 


26.42 


15 


4.77 


38 


12.10 


01 


19.42 


84 


26.74 


10 


5.09 


39 


12.41 


02 


19.74 


85 


27.00 


17 


5.41 


40 


12.73 


03 


20.05 


86 


27.37 


18 


5.73 


41 


13.05 


04 


20.37 


87 


27.09 


19 


0.05 


42 


13.37 


05 


20.09 


88 


28.01 


20 


6.37 


43 


13.09 


00 


21.01 


89 


28.33 


21 


0.68 . 


44 


14.00 


07 


21.33 


90 


28.05 


22 


7.00 


45 


14.32 


08 


21.05 


91 


28.97 


2;] 


732 


46 


14.04 


09 


21.96 


92 


29.28 


24 


7.64 


47 


14.96 


70 


22.28 


93 


29.00 


25 


7.96 


48 


15.28 


71 


22.60 


94 


29.92 


26 


8.28 


49 


15.00 


72 


22.92 


95 


30.24 


27 


8.59 


50 


15.92 


73 


23.24 


96 


30.50 


28 


8.91 


51 


10.23 


74 


23.55 


97 


30.88 


29 


9.23 


52 


16.55 


75 


23.87 


98 


31.19 


30 


9.55 


53 


16.87 


70 


24.19 


99 


31.51 


31 


9.87 


54 


17.19 


77 


24.51 


100 


31.83 



This table is used in the following manner : — 
1. Given the circular pitch and the number of teeth, to 
find the pitch diameter. Take from the table the diameter 



8^ ELEMENTARY MECHANISM. 

corresponding to the given number of teeth, and multiply this 
tabular diameter by the given pitch in inches. The product 
will be the required pitch diameter in inches. 

2. Given the pitch diameter and the number of teeth, to 
find the pitch. Take from the table the diameter correspond- 
ing to the given number of teeth, and divide the given pitch 
diameter by this tabular diameter. The quotient will be the 
required pitch in inches. 

3. Given the pitch and the pitch diameter, to find the 
number of teeth. Divide the pitch diameter by the pitch ; 
and, taking the quotient as a tabular pitch diameter, find from 
the table the number of teeth corresponding to this tabular 
diameter. If the latter is not found in the table, the pitch 
assumed is not an aliquot part of the pitch circumference, 
and must be altered slightly so as to agree with the number 
of teeth corresponding to either the next larger or next 
smaller tabular diameter. 

88. Diametral Pitch. — It has been shown in the last 
article, that the relation between the circular pitch, the pitch 
diameter, and the number of teeth, introduces the incon- 
venient number 3.1416. As the number of teeth must be an 
integer, and as the pitch is usually taken some convenient 
part of an inch, it follows that the pitch diameter will very 
often contain an awkward decimal fraction. This may be 
obviated by the use of the diametral pitch, which is being 
rapidly introduced in this country. 

As the circular pitch is obtained by dividing the pitch 
circumference by the number of teeth, so another ratio may 
be obtained by dividing the pitch diameter by the number of 
teeth. In practice, it is found more convenient to invert this 
last ratio ; and, when so inverted, it is called the diametral 
pitch, though theoretically that designation would more prop- 
erly belong to the ratio as it stood before inversion. In other 
words, we define diametral j)itch to be the number of teeth 
per inch of pitch diameter. Thus, a wheel which has 8 teeth 



MOTION BY SLIDING CONTACT. 



83 



per inch of pitch diameter, is spoken of as an " 8-pitch " 
wheel. 

The chief merit of this system, and one which entitles it to 
great favor, is, that it establishes a convenient and manage- 
able relation between the pitch diameter and the number of 
teeth ; so that the calculations are of the simplest descrip- 
tion, and the results convenient and accurate. 

Let M = diametral pitch ; then we have MP = 3.1416, or 
the product of the circular and the diametral pitches is the 
number 3.1416. 

In this system, the number of teeth and the pitch diameter 
are so related that the circular pitch is usually some decimal ; 
but this is of slight importance, as the circular pitch is rarely 
set off by actual measurement, but usually by dividing the 
pitch circle into the required number of parts. 

To find the number of teeth in any wheel, multiply the 
diametral pitch by the pitch diameter. For instance, an 
8-pitch wheel of 12 inches pitch diameter has 8 x 12 = 96 
teeth. 

Again : to find the pitch diameter, divide the number of 
teeth by the pitch. Thus, a 6-pitch wheel of 25 teeth has a 
pitch diameter of ^^- = 4 J inches. 

In the comparison of circular and diametral pitches, the 
following table will be found useful : — 



A 


B 


A 


B 


A 


B 


A 


B 


1 

4 


12.56 


If 


1.80 


3i 


0.90 


7 


0.45 


i 


6.28 


2 


1.57 


4 


0.78 


8 


0.39 


f 


4.20 


2i 


1.40 


4i 


0.70 


9 


0.35 


1 


3.14 


2i 


1.25 


5 


0.63 


10 


0.31 


u 


2.50 


2f 


1.15 


51 


0.58 


12 


0.26 


li 


2.10 


3 


1.05 


6 


0.52 


16 


0.20 



Find the given pitch, circular or diametral as the case may 



84 ELEMENTARY MECHANISM. 

be, in column A ; then the equivalent pitch in the other sys- 
tem will be found opposite in column B. 

In this volume, circular pitch is always meant when the 
word " pitch " is used without further qualification. 



MOTION BY SLIDING CONTACT. 85 



CHAPTER VI. 

COMMUNICATION OF MOTION BY SLIDING CONTACT. 

VELOCITY RATIO CONSTANT. 

DIRECTIONAL RELATION CONSTANT. 

TEETH OF WHEELS (CONTINUED). 

Definitions. — Angle and Arc of Action.— Epicydoidal System.— 
Interchangeable Wheels. — Annular Wheels. — Customary Dimen- 
sions. — Involute System. 

89. Teetli. Definitions. — That part of the front or 
acting surface of a tooth which projects beyond the pitch 
surface is called the face, and that part which lies within the 
pitch surface is called the Jlank. The corresponding portions 
of the back of a tooth may be called the back face and the 
back jlank. The face of a tooth in outside gearing is always 
convex ; ihQ flank may be convex, plane, or concave. By the 
pitch point of a tooth is meant the point where the pitch line 
cuts the front of the tooth. In Fig. 72, let the front or 
acting surface of the teeth be to the left. Then 6, A;, are 
the pitch points of the teeth ; ab is the /ace; bm is th& flank; 
de is the back face; en is the back flank. 

The depth, AD, of a tooth is the radial distance from root 
to top ; that portion of the top of a tooth which projects be- 
yond the pitch surface is called the addendum, AB ; and a 
line drawn parallel to the pitch line, and touching the tops of 
all the teeth of a wheel or rack, is called the addendum line, 
or, in circular wheels, the addendum circle, adA. The radius 



86 ELEMENTARY MECHANISM. 

of the pitch circle of a circular wheel is called the geometrical 
or pitch radius ; that of the addendum circle is called the real 
radius ; their difference is evidently the addendum. 

Clearance is the excess of the total depth above the work- 
ing depth ; or, in other words, the least distance between the 
top of the tooth of one wheel and the bottom of the space 
between two teeth of another wheel, with which the first 
wheel gears. 

Backla&h is the excess of the space between the teeth of 
one wheel over the thickness of the teeth of another wheel, 
with which the first wheel gears. The amount of backlash 
depends on the accuracy with which the teeth are constructed, 
and should always be made as small as possible. For our 
present purposes we may neglect it altogether. 

90. Ang-le and Arc of Action. — The angle through 
which a wheel turns, from the time when one of its teeth 
comes in contact with the en2:aoino- tooth of another wheel 
until their point of contact has reached the line of centres, is 
called the angle of ajyproacJi; the angle through which it 
turns from the instant that the point of contact leaves the 
line of centres until the teeth quit contact, is called the a7igle 
of recess. The sum of these two angles is called the angle of 
action. The arcs of the pitch circles which measure these 
angles are called the arcs of approach^ recess, and action 
respectively. The corresponding arcs must evidently be the 
same in both pitch circles, while the corresponding angles 
are proportional to the velocity ratio ; in other words, in- 
versely proportional to the diameters of the pitch circles. 

In order that one pair of teeth may continue in contact 
until the next pair begin to act, the arc of action must be at 
least equal to the pitch arc, and in practice it ought to be 
considerably greater. 

Now, in practice, the friction which takes place between 
surfaces whose points of contact are approaching the line of 
centres is found to be of a much more vibratory and injurious 



MOTION BY SLIDING CONTACT. 



87 



character than that which takes place while the points of 
contact are receding from the line of centres. It is therefore 
expedient to avoid the first kind of contact as much as 
possible. 

91. Construction of Tooth Outlines. — In Fig. Go, 
let A and B be the centres of driving and following wlieels 
respectivel3\ Let T be found as usual. Draw the pitch 




3^g.6a 



88 . ELEMENTARY MECHANISM. 

circle MN and RS, and assume a describing circle of any 

T)/T7 

radius OT, less than — -. Let Ta, Th, be the given pitch 

arcs, and lay off on the describing circle the arc TP =z Ta = 
Th. If we now roll this describing circle on the outside of 
MN^ the point P will describe the epicycloid Pa; and this 
cu"ve will be the face of the driver's tooth. Bisect Ta at H^ 
and draw the epicycloid Hp^ similar to Pa, but reversed in 
position. Join pP by a circular arc concentric with MN; 
then HpPa will be the complete outline of that part of the 
driver's tooth which projects beyond the pitch line. 

If we now roll the same describing circle on the inside of 
RS^ the point P will describe the hypocycloid P&, which will 
be the acting flank of the follower. But although Ph is all 
of the flank of the follower's tooth that comes in contact with 
the face Pa of the driver's tooth, yet in order to make room 
for the point of the latter, as it revolves, it is necessary to 
lengthen the driver's flank. This is usually done by con- 
tinuing the hypocycloid hP to i>, making the depth of the 
follower's teeth within the pitch circle PS slightly greater 
than the height of the driver's teeth beyond the pitch circle 
MN. 

The bottom of the space between the follower's flanks con- 
sists of a circular arc concentric with RS. 

To determine the faces of the follower's teeth and the 
flanks of the driver's teeth, we proceed in a precisely similar 
manner. Assuming a describing circle with radius TL less 

AT 

than -^, and rolling the same on the outside of RS^ the 

point P' will describe the epicycloid P'h' for the face of the 
follower's teeth. Again, rolling the same circle on the inside 
of IfA^, the point P' describes the acting flank dP' of the 
driver, which must be extended to G^', as in the case of the 
follower's flanks. By laying off half the pitch arc around 
the circumferences of both pitch circles, and drawing through 



MOTION BY SLIDING CONTACT. 89 

these points curves similar to tliose already found, but alter- 
nately reversed in position, and terminating them at the top 
of the faces and bottom of the flanks by circular arcs con- 
centric with the pitch circles, we will obtain the complete 
tooth outlines for both wheels. 

92. In this construction, the driver's flank first comes 
into contact with the follower's face at P\ The driver mov- 
ing as indicated by the arrow, the point of contact travels 
along the lower describing circle in the arc P'T, until it 
reaches 2", where the action between the driver's flank and 
the follower's face ceases, and that between the driver's face 
and the follower's flank begins. 

The driver still moving as indicated, the point of contact 
travels along the arc TP of the upper describing circle, and 
at P the contact ceases. 

The points P' and P may be assumed at pleasure on the 
circumferences of the respective describing circles., and will 
fix the lengths of the arcs of approach and recess. In the 
figure, they have been so chosen as to give an arc of approach 
and an arc of recess each equal to the pitch. These arcs are 
usually made equal if each wheel is to act indiscriminately as 
driver or follower ; but if the same w^heel is always to drive, 
the arc of recess, for the sake of freedom from vibratory 
motion, is usually made the greater. 

The arc of approach evidently governs the length of face 
of the follower's tooth, and the arc of recess the length of 
face of the driver's tooth. 

93. Draw the radial line AP. and let K be the intersection 
of AP with the pitch circle MN. As pointed out by Pro- 
fessor Willis, Ka may be equal to, but can never be greater 
than, half the thickness of the tooth, as required by the pitch. 

In the figure, Ka is less than half the thickness of the tooth. 
Had the point P been so taken that Ka had been just half 
this thickness, the tooth of the driver would evidently have 
been pointed. 



90 ELEMENTARY MECHANISM. 

94. Size of Describing Circlec — The lengths and 
shapes of the faces and flanks of the teeth of the wheels, 
with given arcs of approach and recess, evidently depend 
on the relation between the diameters of the pitch and de- 
scribing circles. 

If, in Fig. 65, the diameter of the upper describing circle 
were increased, the face Pa would become shorter, and the 
curvature of both Pa and Ph would decrease, until, when 
the diameter of the describing circle became just equal to 
the radius of RS^ the hypocycloid Ph would become a straight 
line passing through the centre of the pitch circle RS (Art. 
75). This fact is often taken advantage of in laying out 
teeth. When the diameters of both describing circles are 
thus taken equal to the radii of the pitch circle in which they 
roll, the flanks of the teeth of both wheels become radial 
lines, while the faces remain epicycloids. The consequent 
reduction in the labor of laying out the shape of such teeth 
has led to their extensive introduction ; though, in conse- 
quence of the convergence of their radial flanks, they have 
the disadvantage of being comparatively weak at the root. 
If the diameter of the describing circle be made still larger, 
the hypocycloidal flanks will converge still more as they 
recede from the pitch circle, making the tooth still weaker at 
the root. Though describing circles have been successfully 
used having a diameter five-eighths as great as that of the 
pitch circle in which they roll, yet it seems a good practical 
rule to make the radial flank the limit in this direction. The 
smaller the describing circles, the longer will be the faces of 
the teeth, and the greater will be the consequent obliquity of 
action ; but, on the other hand, the stronger will be the tooth. 
"We thus have the two conflicting conditions of obliquity of 
action and strength of teeth, and the size of the describing 
circle will be regulated in each case by their relative impor- 
tance. A good general rule, which is found to work well in 
practice, is to make each describing circle of a diameter 



MOTION BY SLIDING CONTACT. 



91 



equal to tliree-eighths of the diameter of the pitch circle in 
which it rolls. 

95. Relation between Pitcli and Arcs of Approach 

and Recess. — The diameters of pitch and describing circles 
)3eing given, and certain arcs of approach and recess being 
required, to determine the limits between which the pitch 
may var3^ 

B 




In Fig. 66, let MN, RS, be the pitch circles, and CT the 
radius of the upper describing circle. Lay off Ta — Th, the 
arc of recess desired. Lay off the arc TP= Tb, thus fixing 
the position of P. Describe the epicycloid Fa, and draw 



92 ELEMENTARY MECHANISM. 

PA, cutting 3/iV in K. Now, as previously explained, if Ka 
is equal to or less than half the thickness of the tooth, — in 
other words, if Ka is equal to or less than one-fourth the 
pitch, — the construction is possible. Hence the pitch of 
the teeth of the driver must be equal to or greater than four 
times Ka, If it is just equal to four times Ka, the teeth will 
be pointed ; if greater, they will have some thickness at the 
top. 

Let Ta' = Th' be the given arc of approach ; then, by a 
similar construction, we find that the pitch of the teeth of the 
follower must be equal to or greater than four times Kh\ 

Agam : it is evident that the pitch of the driver's teeth 
cannot be greater than the arc aa' ; for, if it were, one pair of 
teeth would quit contact at P before the next pair would come 
into contact at P\ Similarly, the pitch of the follower's 
teeth cannot be greater than the arc hV. But aa' = hh' = 
total arc of action. The pitch of the teeth of both wheels 
must evidently be the same ; hence we find, that, to secure 
the desired arcs of approach and recess, the pitch must not 
he greater than the total arc of action, nor less than either 
4Ka or UiV. 

The pitch being given, to find the arcs of approach and 
recess, draw a radius of MN, and lay off on MN, from the 
point where the radius intersects the latter, an arc = ^ pitch. 
Through the point so found draw the epicycloid which would 
be formed by rolling the describing circle 02' on MN, until 
it meets and intersects the radius at some point. Through 
this point of intersection draw a circular arc concentric with 
MN', where the latter cuts the describing circle will be the 
point P, and the arc of recess will be determined on the sup- 
position that the teeth are pointed. If they are not pointed, 
let X be the addendum ; then a circular arc with radius AT -{- 
X will cut the describing circle at the point of quitting con- 
tact, P, as before. 

Th2 arc of approach is found in a similar manner. 



MOTION BY SLIDING CONTACT. 93 

For example, let the pitch aud describmg circles be given 
as in Fig. 66, and let the required arcs of approach and 
recess be J inch and | inch respectively. Lay off TP = Ta 
=zTh = l ii^ch, aud TP' = Ta' ^ Tb' = ^ inch. Drawing 
the radial lines AP and BP', we find that Ka ~ \\ inch, and 
K'h' — -^^ inch. Hence the pitch cannot be greater than 
-| + 1 = 1|- inches, nor less than 4/i'a = \^ inch. Both of 
these limiting values of the pitch are, however, to be avoided 
in practice. For, if the pitch be taken at its smallest pos- 
sible value, the teeth of the driver will be pointed, and with 
any wear at the points, the desired arc of recess will no 
longer be secured ; while, on the other hand, if the maximum 
possible value be given to the pitch, the action will not be 
smooth, as only one pair of teeth will be in gear at the same 
time. In addition, the possible values of the pitch will be 
further limited by the fact that the pitch must be an aliquot 
part of both pitch circumferences. 

Again, let the pitch be given at one inch, and let it be 
required to determine the maximum arcs of approach and 
recess. Draw the radius Ad and lay off mn = J pitch = J 
inch. Draw the epicycloid nt, and through t describe a cir- 
cular arc tjJ concentric with MN; then Tp = 0.79 inch is the 
maximum arc of recess. Proceeding similarly, we find that 
Tp' = 0.81 inch is the maximum arc of approach. But these 
arcs are determined on the supposition that the teeth of both 
wheels are pointed. In any practical ease, somewhat smaller 
arcs should be used, so as to give the teeth some thickness 
at the top. 

96. Wheels having- Arcs of Recess only. — As pre- 
viously pointed out, the arc of approach depends on the length 
of face of the follower's tooth. But from the considerations 
concerning friction (Art. 90) , it is evident that where a very 
smooth action is required, the arc of approach is objectionable ; 
and in such cases it may be gotten rid of altogether by the simple 
expedient of cutting off the follower's teeth at the pitch circle. 



94 



ELEMENTARY MECHANISM. 



The follower's teeth, then, having no faces, of course the 
driver's teeth will need no flanks. In Fig. 67 is shown the 
construction of a pair of wheels of this kind. The diagram 
is drawn full size, and is the practical solution of the folio w« 
mg problem : — 




Fig. 67 



Distance between centres of pitch circles, 9 inches. Driver 

(lower wheel) to have 40 teeth ; follower, 50 teeth. Arc of 

, recess = IJ times the pitch. Divide line of centres AB at T 

so that — - i^ — ^ — = -. Hence the radius of the pitch 
BT a 50 5 ^ 

circle MJSf = 4 inches, and that of the pitch circle MS is 5 

inches. 

Let the driver move as indicated by the arrow. Take the 

diameter of the describing circle = f of that of the pitch 

circle ES of the follower = f X 10 = 3| inches. Find the 



MOTION BY SLIDING CONTACT. 9^ 

pitch b}^ dividing the circumference of MN into 40 equal 
parts, and lay off the arc Ta = 1^ x the pitch so found. Laji 
off the arcs TP = Th = Ta, also Ha = ^ pitch. Roll the 
describing circle on the outside of 3IJyf and on the inside of 
IiS, describing the epicycloid Pa and the hypocycloid Pb 
respectivel3\ Drawing a radial line from P to the centre of 
My, we find Ka to be less than ^ Ha ; hence the case is a 
practicable one. 

Through H draw an epicycloid HE similar to Pa, but 
reversed in position ; through P draw an arc of a circle PE 
concentric with My, and cutting HE at E. Lay off bF = 
Ha, through F draw a reverse hypocycloid similar to Pb, 
and join F and b by an arc of the pitch circle PS. Now, Pb 
is all of the hypocycloid that comes into contact with the epi- 
cycloid Pa ; but, in order to provide room for the point of the 
latter, the hypocycloid is continued to D, just as was done in 
Fig. 65. 

If the workmanship were accurate, the wheels would work 
properly, provided the depth of the space between two suc- 
cessive teeth of one wheel were just equal to the height of 
the teeth of the other. To provide against any accidental 
contact, however, both sets of teeth are given clearance; that 
is, the bottoms of the spaces between the teeth are formed 
by arcs of circles concentric with MN and RS respectively, 
and at such a distance as to leave a clearance of about one- 
tenth the pitch in both wheels. The outlines of the teeth 
are then completed by joining the bottoms of the epicycloids 
and hypocycloid previously drawn, to these arcs by means of 
small fillets, as shown in the figure. 

The teeth will come into contact at T, the point of contact 
travelling in the arc TP, until it reaches the point P, where 
the contact ceases. It is evident that, before any one pair 
quits contact at P, another pair will have been in contact 
while the wheels were moving over one-third the arc of 
action. 



96 



ELEMENTARY MECHANISM. 



97. Wheels with Arcs of Approach and of Recess. — 

Wheels such as shown in ¥\g. G7 are sometimes used to great 
advantage, particularly in light mechanism where smoothness 
of action is especially important. But whenever the pressure 
to be transmitted is at all heavy, the wheels should have arcs 
of both approach and recess, so that more teeth may be in 
action at the same time. By this means the pressure is dis- 
tributed over more teeth, while the maximum obliquity of the 
line of action is not increased. This is, in fact, the form 
most usually employed in practice, and in Fig. 68 is shown 
the method of laying out a pair of such wheels. The diagram 




is drawn full size, and is the practical solution of the follow- 
ing problem : Distance between centres to be 9 inches. The 
driver (lower wheel) to have 40 teeth, and the follower 50 
teeth. Arc of approach to be equal to the pitch, and the aro 



MOTION BY SLIDING CONTACT. 97 

of recess to be one and a half times the pitch. The condi- 
tions given are the same as those given in Art. 96, except 
that there is to be an arc of approach in this case. The 
pitch radii of the wheels are 4 and 5 inches, as before ; and 
the diameters of the respective describing circles are 3 and 
3f inches. 

The faces of the driver's teeth and the flanks of the fol- 
lower's teeth are found as in Art. 96, and are, in fact, iden- 
tical with those there found. In this case, however, we do 
not finish off the bottoms of the faces of the driver's teeth 
and the tops of the flanks of the follower's teeth by arcs of 
circles, as is done in Fig. 67. 

Lay off the arc TP^ = TV — arc of approach. Using the 
describing circle of three inches diameter, and going through 
the process explained in Art. 91, we obtain flanks for the 
driver's teeth, and faces for those of the follower. By this 
construction, as shown in Fig. ^'^^ there are three pair of 
teeth in contact ; one just quitting contact at P, another in 
contact at _p, and a third pair at jf . In practice, after we 
have determined that the given arcs of action may be secured 
with the given pitch (Art. 95), the four curves are usually 
laid down at T, as shown {Td and Tcj being epicycloids, and 
Te and Tli hypocycloids) . The addendum circle bounding 
the tops of the teeth, and the root circle bounding the bottoms 
of the spaces, are next drawn. The pitch points of the teeth 
are then laid off on the respective pitch circles, and the re- 
spective curves are drawn through the successive pitch points 
in alternately reversed directions, being limited at the top by 
the addendum circle, and connected at the bottom by fillets 
to the arcs of the root circle. 

98. Interchangeable "Wlieels. — If the describing circle 
be made of a diameter bearing a fixed ratio to that of the 
pitch circle, any pair of wheels so laid out will work together ; 
but they cannot both work properly with a third wheel of 
different diameter. Thus, a given wheel having radial flanks 



98 ELEMENTARY MECHANISM. 

cannot work properly with tvjo or more other wheels of dif- 
ferent diameters, and also having radial flanks. 

If, however, we use the same describing circle for all the 
faces and all the flanks, we will obtain a series of inter- 
changeable wheels, any one of which will work correctly with 
any other of the same set. This suggestion is due to Pro- 
fessor Willis, and this method of laying out teeth is invaluable 
for such purposes as constructing the change- wheels of a lathe. 

As, with a constant describing circle, the outlines of the 
teeth will vary with the diameters of the wheels, so as to 
make the obliquity of action greater as the latter increases, it 
is usually advisable to employ as large a describing circle as 
possible. From the considerations discussed in Art. 95, the 
practical rule follows, that, for a set of interchangeable 
wheels, the diameter of the constant describing circle should 
be half the diameter of the pitch circle of the smallest wheel 
of the set. 

99. Rack and Wheel. — When a wheel works with a 
rack^ the line of centres becomes a perpendicular to the pitch 
line of the rack, and passing through the centre of the wheel. 
The rack will travel through a distance equal to the circum- 
ference of the pitch circle of the wheel for each revolution 
of the latter, whatever the number of teeth. The pitch of 
the rack teeth, therefore, is found by rectifying the pitch arc 
of the wheel, and laying off this rectified arc upon the pitch 
line of the rack. In Fig. 69 the two descri]:»ing circles are 
made of the same diameter, so that any other wheel of the 
same pitch whose tooth outlines are formed by means of 
the same describing circle will also gear with the rack. In 
fact, the rack is merely a special case of the wheel ; and all 
the deductions of the previous articles as to tooth outlines, 
arcs of action, etc., apply, with obvious modifications, to this 
case as well. Both faces and flanks of the rack teeth are 
cycloids (Art. 85) : their tops and bottoms are straight lines. 
The clearance is obtained as usual. 



MOTION P.Y SLIDING CONTACT. 



99 



In the figure, which is drawn full size, the diameter of 
pitch circle of the wheel is four inches, and the wheel has 
forty teeth. The ares of approach and recess are each made 




equal to the pitch. Assuming the rack to drive to the right, 
the contact begins at P\ the point of contact travelling along 
the arcs P'T and TP\ and at P the action ends. 



100 



ELEMENTARY MECHANISM. 



The principle of making teeth with straight flanks may, of 
course, be extended to the case of a rack and wheel, as shown 
in Fig. 70. The describing circle whose diameter is TB^ the 
radius of the wheel, generates the cydoidal faces of the rack 
teeth and the radial flanks of the ivheel teeth. The radius of 
the rack being infinite, the diameter of the other describing 
circle is also infinite ; in other words, it is a straight line. 




inig, 70 



Hence the faces of the ivheel teeth are evidently involutes of 
the pitch circle, while the flanks of the rack teeth are straight 
lines perpendicular to TIOT. The arcs of action and the 
addendum of the rack teeth are found as before. The rack 
driving to the right, the contact begins at P' (the point of 
intersection of the wheel addendum circle with the line MN) , 
travels along the straight line P' T, then along the arc TP to 
the point P (the intersection of the rack addendum line with 
the describing circle) , where the teeth quit contact. In this 
form of rack tooth the acting flank has degenerated into a 
mere point, which is consequently subjected to excessive 



MOTION BY SLIDING CONTACT. 



101 



wear. This is a serious defect, and forms a grave objection 
to the use of this form of tooth for racks. 

100. Aiinular Wheels. — The construction explained 
in Art. 91 is applicable not only to tlie case of wheels in 
external gear, as there shown, but to that of wheels in iuter- 

^A 




Fig. ♦71 

nal gear as well. Fig. 71 is drawn full size, and is a prac- 
tical solution of the following problem : Distance between 
centres of pitch circles, 3 inches. The pinion to be the 
driver, and to have 20 teeth ; the annular wheel to have 
50 teeth. The arc of approach and the arc of recess to be 
each equal to the pitch. The radii of the pitch circles of the 
two wheels are evidently 2 and 5 inches respectively. As- 
suming the diameters of tlie respective describing circles at IJ 
and 3f inches, we proceed with the construction as before. 
In fact, on comparing this diagram with Fig. 68, both figures 



102 ELEMENTARY MECHANISM. 

being similarly lettered, we will see that all the details of 
construction are the same in both. The pinion is an ordinary 
spur wheel ; while the acting curves of tne annular wheel are 
identical with those of a spur wheel, having the same pitch 
and describing circles, the tooth of the one corresponding to 
the space of the other. 

The principle of interchangeability (Art. 98) applies to 
annular wheels just as to spur wheels. Thus, a set of spur 
and annular wheels may be made in which each spur wheel 
will gear not only with every other spur wheel, but also with 
every annular wheel. In this case, however, there must be a 
difference in the number of teeth of the spur and annular 
wheels which are to gear together, at least equal to the num- 
ber of teeth on the smallest pinion of the set. 

101. Customary Dimensions of Teeth. — By the pre- 
ceding methods we may design the teeth of gear wheels so as 
to fulfil any proposed conditions as to the relative amounts of 
approaching and receding action. In the majority of cases, 
however, the precise lengths of the arcs of approach and 
recess are not a matter of importance ; and under these cir- 
cumstances it is customary to make the whole radial height 
of the tooth a certain definite fraction of the pitch, the part 
without the pitch circle being a little less than that within, by 
which clearance is provided for. 

There are a number of such arbitrary proportions ; but 
none of them can be considered absolute, as the proper 
amount of clearance and backlash evidently depends on the 
precision with which the tooth curves are laid out, in the first 
place, and on the accuracy with which the shapes of the teeth 
are made to conform to the curves so found. 

In the manufacture of the best cut gears at the present 
day, the backs of the teeth barely clear each other when the 
fronts are in contact ; but in the majority of cases a greater 
allowance is still made, depending for its amount on the accu- 
racy of the workmanship. In cast wheels backlash is abso- 



MOTION BY SLIDIXG CONTACT. 



103 



hitelv necessary to allow for irregular shrinkage or accidental 
deranoement of the mould. 




Fig. 7% 



In Fig. 72, let hk — circular pitch = P. Then, accord- 
ing to several systems in general use for proportioning teeth, 
we have the following values : — 



Total depth . . AD 


-fo-P 


0.75P 


MP 


O.750P 


Clearance . . . C7) 


-A-P 


0.05P 


i.-P 


0.060P + 0.04 in. 


Working-depth . AC 


i%P 


0.70P 


HP 


0.690P - 0.04 in. 


AC 

Addendum, AB — ^ 


'hP 


0.35P 


S- 


0.345P — 0.02 in. 


Thickness of tooth, he 


AP 


0.45P 


'hP 


0.470P - 0.02 in. 


Width of space . ke 


r^-P 


0.55P 


AP 


0.530P + 0.02 in. 


Backlash . . ke — he 


-hP 


O.IOP 


-hP 


0.060P + 0.04 in. 



In the first three systems the percentage of backlash is 
constant, the actual amount of backlash thus increasing 
directly with the pitch. It seems more rational, however, to 
make the percentage of backlash greater for small pitches 
than for large ones ; for, the coarser the pitch, the smaller 
will be the proportion borne to it by any unavoidable error. 
The last s^^stera, that of Fairbairn and Rankine, is founded 
on this view of the proper proportion of backlash. In this 



104 ELEMENTARY MECHANISM. 

s^^stem the percentage of backlash gradually diminishes as 
the pitch increases. The actual amount as given by this 
system is, however, rather larger than is generally used at 
present. Teeth proportioned by any of these systems will 
in general be of good shape, and answer the purpose desired. 
Should the wheel have less than about twelve teeth, or should 
the exact amount of approaching or receding action be of 
importance, no arbitrary system should be used. In all such 
cases the proper dimensions of the teeth should be found as 
previously explained. The backlash and clearance should 
always be made as small as the character of the workmanship 
will permit. In our diagrams we have assumed no backlash 
to exist ; but its introduction would have no effect, except to 
diminish the thickness of the tooth. Instead of half the pitch, 
as it is in the diagrams, the thickness of the tooth would be 
half the pitch minus half the backlash. In using the diametral 
pitch, the working depth of a tooth is almost always taken at 
two pitch parts of an inch, and the addendum at one pitch 
part of an inch. That is, in a 4-pitch wheel, the working 
depth is f = |- inch, and the addendum is i inch. The clear- 
ance and backlash are taken at from a fourth to an eighth of 
one pitch part of an inch ;" thus, in a 4-pitch wheel, they would 

be taken at from ( = — ] to f = — ) of an inch. 

4 X 4\ 16/ 8 X 4V 32/ 

The simplicity of these proportions have led to their almost 

universal adoption whenever the diametral pitch is employed. 

102. Involute System. — It has been shown (Art. 77) 
that involutes of certain circles possess the property of trans- 
mitting motion by sliding contact with a constant velocity 
ratio, and the application of such curves to the formation of 
the teeth of wheels is shown in Fig. 73. 

Let AB be the line of centres, divided at T, so that 

AT a 

— — = — . Draw the pitch circles MN and BS, and their 
ST a 

common tangent t'Tt. Draw j/T^), making an oblique angle 



MOTION BY SLIDING CONTACT. 105" 

pTt with the tangent t'Tt. From the centres A and B drop 
the perpendiculars ^p' and Bp on the line p'Tp ; and with 
these perpendiculars as radii, describe the circles M'N' and 
B!Si\ which will be tangent to the line p'Tp. 

\B 



For the sake of simplicity, let the arcs of approach and of 
recess each equal the pitch. On the pitch circle MN^ lay 



106 ELEMENTARY MECHANISM. 

off the pitch Ta. With M'N' as a base circle, draw the in- 
volute a!'P^ passing through the point a, and intersecting the 
line p'Tp at P. Then o!'P will be the acting outline of the 
driver's tooth ; and, similarly, h"P' will be the acting outline 
of the follower's tooth. The tooth outlines of each wheel 
are evidently continuous curves, there being no marked divis- 
ions into face and flank, as in the epicycloidal system. Com- 
pleting the tooth outlines, as shown in the diagram, we find, 
that, as in the case of epicycloidal teeth, room must be pro- 
vided for the points of the teeth as they revolve. As the 
involutes cannot extend within their own base circles, this 
clearance space is provided by continuing the flanks b}^ radial 
lines, and joining the latter by means of circular arcs. The 
contact begins at P^, and during the action the point of con- 
tact travels along the line iilTp till it reaches P, where contact 
ceases. By making the teeth of both wheels pointed, we can 
evidently cause them to begin contact atp^ and quit contact at p. 

If this is done, each involute will be long enough to touch 
the other at its root, and the arcs of approach and recess will 
be directly proportional to the radii of the pitch circles of 
the driver and follower respectively. Where pointed teeth 
are to be employed, it follows that the action will be 
smoother when the smaller wheel is the driver. But pointed 
teeth are objectionable here just as in the epicycloidal sys- 
tem ; so that, practically, the arcs of approach and recess 
are adjusted for each particular case, by making the teeth of 
the proper length. 

103. Given the pitch circles, the obliquity of the line of 
action, and the desired arcs of approach and recess, to find 
the limiting values of the pitch which will secure these arcs 
of action. The receding action evidently continues while the 
point of contact travels from I^ to P in the line TP^ a dis- 
tance equal to the arc Oc^' . 

The curves OTH and (.("aP being equal involutes of M'N\ 
and the points T, a, lying in the circumference of the circle 
MN^ concentric with that of M'N' which contains the points 



MOTION BY SLIDING CONTACT. 107 

0, o!\ it follows thr.t the angle TAa = angle OAq\!\ and 

arc Oa" A// jj ^ „ Ar/ rn 

= —!—. Hence Oa = —^— x Ta. 

arc Ta AT AT 

On the tangent Tt lay off the distance Td = Ta ; from d 
draw a perpendicular to TP. Then, from the similar trian- 
gles TAW and TdP, we will have TP = ^^ X Ta = Oa" , 

as required. 

Draw tlie radius PA^ cutting MN in K. Now, the pitch 
cannot be less than 4/i'a. If it is just equal to 47ra, the 
teeth will be pointed ; if greater, they will have some thick- 
ness at the top. Similarly, the pitch cannot be less than 
4,K'h. Hence we find that the pitch cannot be greater than 
the total arc of action nor less than either 4/i'a or 'iK'h. 

104. Given the pitch circles, the obliquity of the line of 
action, and the pitch, to find the arcs of approach and recess. 
From T lay off on the circle MN an arc = J pitch, and 
through the point so found draw an involute of the circle 
M'N' . Through the point of intersection of this involute 
with the line AP^ draw a circular arc concentric with 
MN^ and cutting the line p'Tp at some point P. Then P 
will be the point at which the teeth will quit contact, and 

Ta = ^ — ; X TP will be the arc of recess. This is only true 
Ai/ 

if the teeth are pointed ; if they are not, let .i' be the adden- 
dum. Then a circular arc struck about ^1, with radius AT 
4- 0.", will cntp'Tj^ in the point where the teeth quit coutaet. 
The arc of approach is determined in a similar manner. 

105. Practical Example. — Fig. 74 is drawn full size, 
and is the practical solution of the following ])roblem : — 

Distance between centres, 9 inches. Driver (lower wheel) 
to have 40 teeth ; follower, 50 teeth. The constant obliquity 
of the line of action to be 15°. Draw the pitch circles 3IN 
and PS with radii of 4 and 5 inches respectively, and their 
common tangent t'Tt. Draw the line of action DE, making 



108 



ELEMENTARY MECHANISM. 



the angle ETt — 15°. On DE drop the perpendiculars AD 
and BE^ with which, as radii, describe the base circles M'N' 
and BIS', 

The arcs of approach and recess in this problem are each 
to be equal to the pitch. Hence lay off the pitch arc Ta ; 

and lay off, on the line of action, the distance TP = — — x 

AT 

Ta. Then P is the point at which the teeth quit contact. 




TFig. V^ 



As the arcs of approach and recess are to be equal, lay off 
TP' = TP. Then P' is the point at which the teeth first 
come in contact. Through P draw the involute Pa'' of the 
base circle M'N\ and through P' draw the involute P'h" of 
the base circle P'S'. Draw the addendum circles through 
P' and P, lay off the pitch points of the teeth around the 
pitch circles MN and RS^ and draw through the points so 
found, in alternately reversed positions, the involutes Pa" and 
P'h" respectively. 

The tops of the teeth are bounded by arcs of the respective 
addendum circles. To provide clearance, continue the tooth 



MOTION BY SLIDING CONTACT. 109 

outlines from the bottoms of the involutes by radial lines to 
the proper depth. The bottoms of the spaces are circular 
arcs concentric with the centres of motion, and joined to the 
tooth outlines by means of small fillets, as shown. 

In these wheels, there are evidently always two pairs of 
teeth in contact. In the position shown, there is one pair in 
contact at T on the line of centres, while a second pair is 
quitting contact at P at the same moment that a third pair 
is eno^ao-iiioj in contact at P'. 

106. Interference of Involute Teeth. — So long as 
the teeth are of such a length that the points P and P' (Fig. 
74) lie between E and D, they will work properly. In other 
words, the addendum circle of the teeth of the lower wheel 
must lie within a circle through E^ and concentric with MN. 
Also, the addendum circle of the teeth of the upper wheel 
must lie within a circle through i), and concentric with RS. 
But when the dimensions of teeth are decided on by means 
of some arbitrary system, such as those of Art. 101, it fre- 
quently happens that the length of tooth so found will be 
great enough to cause the addendum circles to lie outside of 
the concentric circles through E and D respectively. It fol- 
lows, that the part of the tooth projecting beyond this limiting 
circle will come into contact with that part of the tooth of the 
other wheel which lies within the base circle. As this inner 
part is always made radial, it cannot gear correctly with an 
involute face, and interference will take place. In case a 
tooth of such length is considered necessary, and the involute 
system is to be used, all that part of the face of the tooth 
of one wheel coming into contact with the radial part of the 
tooth of the other wheel must be an epicycloid whose de- 
scribing circle is half the diameter of the pitch circle of the 
second wheel. As, by this means, we forfeit one of the great 
advantages of the involute system (the power of varying the 
distance between centres without affecting the velocity ratio) , 
this construction is not to be recommended, and the length of 



110 ELEMENTARY MECHANISM. 

tooth should not be allowed to exceed the amount determined 
by the methods of the preceding articles. 

107. Rack and Wheel. — If, in Fig. 74, the radius AT 
of the driver were to increase, the curvature of MN, as well 
as that of the involute of M^N\ would necessarily decrease; 
until, when MJSf became a straight line, M'N' would also 
become a straight line, and the involute of M'N' would be- 
come a straight line, which must be perpendicular to the line 
of action, BTE. 



The method of constructing the teeth is exactly similar to 
that shown in Fig. 74. 

In Fig. 75 the rack is the driver, and the follower is tlie 
same wheel that was used as follower in Fig. 74. The teeth 
of the follower remain the same as in the other case, while 
those of the rack have straight sides. The tops and bottoms 
of the rack teeth are straight lines parallel to pitch line MN 
of the rack. In order to drive the follower through one 
complete revolution, the rack will evidently have to travel a 
distance equal to the circumference of the pitch circle of the 
wheel. 

The construction of the teeth of ammlar wheels is also in 
all respects similar to that explained above for spur wheels. 



MOTION BY SLIDING CONTACT. Ill 

108. Peculiar Properties of Involute Teeth. — In 

the preceding constructions of practical problems, the line of 
action was drawn at an angle of 15° with the common tan- 
gent of the two pitch circles. This angle is by no means 
fixed, and may be considerably varied ; but experience has 
shown that for general practice it should 7iot be greater than 
fifteen degrees. As the magnitude of this angle has formed 
no part of the argument in the preceding cases, it follows 
that, by varying the obliquity of action, an infinite number of 
pairs of base circles may be used in connection with any 
given pair of pitch circles. Conversely, with a given pair 
of base circles, we may, by altering the length of the line of 
centres, have an infinite number of pairs of pitch circles. 
The common tangent to the two base circles will always cut 
the line of centres into segments having the same ratio as 
their radii, which will be the same as that of the radii of any 
of the pairs of pitch circles ; from which follow two impor- 
tant practical deductions : — 

1. Any two wheels with involute teeth of which the pitch 
arcs on the base circles are equal, will gear correctly with 
each other. 

2. The velocity ratio will not be affected by any change in 
the distance between their centres. 

The peculiarity of interchangeability is also obtainable 
with epicycloidal teeth under certain conditions (Art. 98). 
The peculiarity of constant velocity ratio with varying dis- 
tance between the centres is not found m any other form of 
teeth, and is of special importance in mechanism requiring 
exceptional smoothness and uniformity of action. The 
shafts may be at the proper distance apart, or not, as 
happens ; and the}^ may change position by wearing, or by 
variable adjustment, as when used on rolls, or they may be 
brought closer together to abolish backlash. In fact, the 
involute tooth is remarkably well adapted to such variable 
demands, and will accommodate itself to errors and defects 
that are difficult to avoid in practice. 



112 ELEMENTARY MECHANISM. 

The line of action of epicycloidal teeth is perpendicular to 
the line of centres at the instant when the point of contact is 
on that line ; but that of involute teeth is constantly in the 
direction of the common tangent of the two base circles, and 
hence always oblique to the line of centres. The obliquity of 
involute teeth, then, is constant ; and it is, in general, greater 
than the mean obliquity of epicycloidal teeth having the same 
angle of action. The thrust on the bearings is therefore 
greater with involute than with epicycloidal teeth ; and 
though for heavy pressures this is* sometimes a serious objec- 
tion to the use of involute teeth, yet for ordinary work it 
would scarcely be so considered. 

The involute tooth has a great advantage over the epicy- 
cloidal tooth in being of a much stronger shape, spreading 
considerably at the root, which in the epicycloidal form is 
often the weakest part. Though the epicycloidal tooth is 
still in much greater use than the involute tooth, yet the 
merits of the latter are being rapidly recognized by manu- 
facturers ; and, for light work at least, it is gradually coming 
into more general use to replace the epicycloidal form. 



MOTION BY SLIDING CONTACT. 113 



CHAPTER VII. 

COMMUNICATION OF MOTION BY SLIDING CONTACT. 

VELOCITY RATIO CONSTANT. 

DIRECTIONAL RELATION CONSTANT. 

TEETH OF WHEELS (CONTINUED). 

Approximate Forms of Teeth. — Willis^ Method. — Willis^ Odonto- 
graph. — Granfs Odontograph. — Bobinson^s Odontograph. 

109. Approximate Forms of Teetli. — In order to 
secure perfect smoothness of action in toothed wheels, it is 
essential that the tooth outlines should be accurately laid out, 
as explained in the preceding pages, and that the teeth 
should be constructed so as to conform exactly with the 
outlines so found. If the teeth are to be cut, there is no 
reason why the exact curves should not be used, for it is 
as easy to form the cutter of the exact shape as of any 
approximate one ; and the cutter once formed, the exact 
curves can be cut as easily as any other. When the teeth 
are to be simply cast, however, or when, for other reasons, 
perfect accuracy is not sought after, we may replace the 
exact curves by others which approximate to them more or 
less closely, but which are simpler to construct. When 
approximate forms of teeth are employed, some one of the 
arbitrary sets of proportions given in Art. 101 is generally 
followed. 

The two principal methods of approximation are by cir- 
cular arcs and by curved templets. 



114 



ELEMENTARY MECHANISM. 



110. Willis' Method of Circular Arcs In Fig. 76, 

let ^ and B be the centres of motion, and T the point of 
contact of the pitch circles MN and RS. Draw the line of 
action DTE, making any assumed angle with AB, smd erect 
on it the perpendicular TO. On TO assume the point 0, 
and through this point draw the lines APO and BOQ. We 
have now formed a system of linkwork, consisting of the 
arms AP and BQ, connected by the link PQ ; and as, by 




IFig. 7Q 



construction, is the instantaneous centre of PQ, it follows 

a AT 

(Art. 25) that — = for that instant. If at any 

^ ^ a BT -^ 

point b on BE we draw two curves, abc and Jibe, in contact, 

and of such shape that P and Q are their respective centres 

of curvature, these curves will, by revolving about centres A 

and B respectively, produce the constant velocity ^ = ~ — , 

a BT 

the same as that of the pitch circles. In the preceding 

articles we have already discussed the theoretical shapes of 

such curves ; and, from the above, it is evident that, if 

circular arcs be drawn through &, with centres P and Q, 



MOTION BY SLIDING CONTACT. 115 

they will fulfil the required condition for that instant. If, 
however, the teeth are short, and the obliquity is not very 
great, these arcs differ so slightly from the true curves that 
they may be substituted for the latter with very good results. 
In the figure the arc ahc will be the f^ce of the tooth of 
MN^ while libe will be the flank of the follower. 

111. Approximate Involute Teeth by Willis' 
Method. — In this case the side of the tooth is made to 
consist of a single arc, and a very simple rule may be 
obtained. 

In Fig. 76, let TO = cc ; then AP and BQ will become 
perpendicular to DE, and the points F and Q will fall at 
F^ and Q^ respectively. 

Let the circular arcs be struck through T; let i? be the 
radius, AT, of the wheel, and <^ the angle which DE makes 
with AB. Then TF^= R cos ^, which is independent of 
the wheel FS, as well as of the pitch and number of teeth 
of MN. If, therefore, the angle <^ be made constant in a 
set of wheels, and their teeth be described by this method, 
any two of them will work together. 

Assume </> = 75° 30', which is a very convenient value, 

for which TF' = R cos 75° 30' = 0.25038i^ = — very 

nearly. 

112. Practical Example. — Let it be required to con- 
struct, by this method, the teeth of a wheel of 25 teeth ; 
diameter of pitch circle, 4 inches. Let AT (=2 inches) be 
the radius (Fig. 77), and MN the pitch circle, of the proposed 
wheel. The pitch, as near as may be, is half an inch. We 
will make the teeth of the proportions given in the first sys- 
tem of Art. 101. This gives addendum = 0.15 inch, total 
depth = 0.35 inch, backlash = 0.04 inch. Hence draw the 
addendum and root circles at distances of 0.15 inch without, 
and 0.20 inch within, the pitch circle, respectively. Draw 
rP, making an angle of 75° 30' with the radius, and drop 



116 



ELEMENTARY MECHANISM. 



a perpendicular, AP, upon TF for describe a semicircle 

upon AT, and set off TF = - — J ; then will F be the centre 

from which an arc, aTh^ described through T, will be the 
side of the tooth required. To describe the other teeth, 
draw, with centre A and radius AF^ a circle, mn, within the 




pitch circle MN\ this will be the locus of the centres for the 
teeth. Set off around the pitch circle, arcs of 0.23 inch and 
0.27 inch in length alternately, being the respective widths 
of tooth and space on the pitch circle. Take the constant 
radius in the compasses, and, keeping one point in the circle 
mn^ step from tooth to tooth, and describe the arcs, as shown 
in the figure, joining them directly to the arcs of the adden- 
dum circle, and by small fillets to tlie arcs of the root circle. 
If aTh were an arc of an involute having mn for a base 
circle, TF would be its radius of curvature at T. These 
teeth, therefore, approximate to involute teeth ; and they 



MOTION BY SLIDING CONTACT. 117 

possess, in common with them, the oblique action, the power 
of acting with wheels of any number of teeth, and the 
adjustment of backlash. But, as the sides of the teeth con- 
sist each of a single arc, there is but one position of action 
in which the angular velocity is strictly constant ; namely, 
when the point of contact is on the line of centres. 

The length of the teeth should always be kept within the 
limits shown in Art. 102, and in such cases the above method 
of approximation will give fairly good results. The larger 
the wheel, the more closely will the circular arcs obtained by 
this rule agree with the true involute curve. 

113. Approximate Epicycloidal Teeth by Willis' 
Metliod. — By making the side of each tooth consist of two 
arcs joined at the pitch circle, and struck in such wise that 
the exact point of action of the one shall lie a little before the 
line of centres, say at the distance of half the pitch, and 
the exact point of the other at the same distance beyond that 
line, an abundant degree of exactitude will be obtained for 
all practical purposes. 

In Fig. 78, let J. and jB be the centres of motion, and 
T the point of contact of the pitch circles JO/" and US. 
Draw DE, making an angle of 75° with AB. This angle 
is, in fact, arbitrary ; but 75° has been found by Professor 
Willis to give the best form to the teeth. 

Draw OTO' perpendicular to DE, and set off the lengths 
TO and T0% equal to each other, and less than either ^T or 
BT. Through draw the lines BOQ and APO, and through 
0' draw the lines BQ'O' and AC/P'. By this construction, 
which is merely an extension of that of Art. 109, we obtain 
four tooth centres. P will be the centre for the faces of 
MN, Q the centre for the flanks of PS, Q for the faces 
of RS, and P' for the flanks of MN. The flayik of RS and 
the face of MN will be circular arcs, with centres Q and P 
respectively, and drawn in contact at a distance of half the 
pitch to the right of the line of centres ; the face of RS and 



118 



ELEMENTARY MECHANISM. 



tlie/a??A; of ilf^ will be circular arcs, with centres Q! and P\ 
and drawn in contact at a distance equal half the pitch to the 
left of the line of centres. 

IB 




From the construction it appears that the teeth of one 
wheel are not changed in shape by any change in the radius 
of the other wheel. In short, if any number of wheels be 
described in the above manner, in which the angle DTA is 
constant, the distances TO and TO' being the same for the 
whole set of wheels, then any two of these wheels will work 
together. The distance TO' may be determined for a set of 
wheels by considering that if A approach T, the point 0' 
remaining fixed, AP' becomes parallel to DE^ and the flank 
of the tooth of MN becomes a straight line. If A approach 
still nearer, P' appears on the opposite side of T, and the 
flank becomes convex, giving a very awkward form to the 
tooth. The greatest value, therefore, that can be given to 
TO and TO' must be one which, when employed with the 



MOTION BY SLIDING CONTACT. 110 

smallest radius of the set, will make AP^ parallel to DE. 
By assuming constant values for this smallest radius, as well 
as for the angle DTA, in a set of wheels, the values of the 
radii of curvature of the faces and flanks which correspond 
to different numbers and pitches, maj^l^e calculated and tabu- 
lated for use, so as to supersede the necessity of making the 
construction in every case. Thus, the values in the tables of 
Fig. 79 were obtained by assuming that the least radius was 
just great enough to give the wheel twelve teeth of the required 
pitch, and that the angle DTA wa« 75°. 

114. Willis' Oclontograpli. — This instrument, repre- 
sented in Fig. 79, was contrived by Professor Willis for the 
purpose of laying out the approximate forms of teeth accord- 
ing to the principles of Art. 113. The figure represents the 
instrument exactly half the size of the original ; but, as it 
may be made of a sheet of bristol-board, this figure will 
enable any one to make it for use. The side NTM^ which 
corresponds to the line DE in Fig. 78, is straight; and the 
line TC makes an angle of exactly 75° with it, and corre- 
sponds to the radius ^2" of the wheel. This side, JSfTM, 
is graduated into a scale of twentieths of inches ; and each 
tenth division is numbered, both ways, from T. 

The instrument is often made of brass, and in that case 
is of the shape shown in Fig. 80 ; the tables not being on the 
instrument, but on a printed sheet accompanying the same. 

The manner of usinoj the instrument is shown in Fis;. 80. 
Let it be required to describe the form of a tooth for a wheel 
of 29 teeth of 3 inches pitch. This determines the radius 
^T of the pitch circle MN. Lay off the arcs TD and TE, 
each equal to half the pitch, and draw the radial lines AD, 
AE. To draw the flank, apply the instrument with its slant 
edge on AD, so that D is at the zero point of the scales. In 
the table headed "Centres for the Flanks of Teeth," look 
down the column of 3-inch pitch, and opposite to 30 teeth, 
which is the nearest number to that required, will be found 



TABLE SHOWING THE PLACE OF THE 


CENTRES UPON THE SCALES. 


CENTRES FOR THE FLANKS OF TEETH. 


NUMBER 

OF 
TEETH. 


PITCH IN INCHES. 


1 


m 


IJ^ 


IM 


2 


2V4 


2% 


3 


13 


129 


160 


193 


225 


257 


289 


321 


386 


14 


69 


87 


104 


121 


139 


156 


173 


208 


15 


49 


62 


74 


86 


99 


111 


123 


148 


16 


40 


50 


59 


69 


79 


89 


99 


12] 


17 


34 


42 


50 


59 


67 


75 


84 


101 


18 


30 


37 


45 


52 


59 


67 


74 


89 


20 


25 


31 


37 


43 


49 


56 


62 


74 


22 


23 


27 


33 


39 


43 


49 


54 


65 


24 


20 


25 


30 


35 


40 


45 


49 


59 


26 


18 


23 


27 


32 


37 


41 


46 


55 


30 


17 


31 


25 


29 


33 


37 


41 


49 


40 


15 


18 


21 


25 


28 


33 


35 


42 


60 


13 


15 


19 


22 


25 


28 


31 


37 


80 


12 


15 


17 


20 


23 


26 


29 


35 


100 


11 


14 


17 


20 


22 


25 


28 


34 


150 


11 


13 


16 


19 


21 


24 


27 


32 


Eack 


10 


12 


15 


17 


20 


22 


25 


30 


CENTRES FOR THE FACES OF TEETH. 


12 


5 


6 


7 


9 


■ 10 


11 


12 


15 


15 


5 


7 


8 


10 


11 


12 


14 


17 


20 


6 


8 


9 


U 


13 


14 


15 


18 


30 


7 


9 


10 


12 


14 


16 


18 


21 


40 


8 


9 


11 


13 


15 


17 


19 


23 


60 


8 


10 


12 


14 


16 


18 


20 


25 


80 


9 


11 


13 


15 


17 


19 


21 


26 


100 


9 


11 


13 


15 


18 


20 


22 


26 


150 


9 


11 


14 


16 


19 


21 


23 


27 


Eack 


10 


12 


15 


17 


20 


23 


25 


30 



WILLIS' ODONTOGRAPH, 





•200- 


-j 




190- 


-| 




180^ 


(fi 


170- 


--f 


o 




-$; 


> 

r 


m--$ 


n 




-^ 


O 


150- 


— ^ 


O 




:$ 


m 


140^ 


^ 


2 




— t$ 




130- 


^ 


m 




5= 


w 


120- 


-^ 


■n 




-^ 


O 


110^ 


-| 


H 




'^ 


X 


100- 


-^ 


m 




-^ 




90- 


-^ 


80- 


J 


U) 




^ 


o 


70- 


-$ 


■n 




^:=: 




60" 


-f 


n 




^ 


H 


50- 


-— =; 


X 




-^ 




40- 


— ^ 




30- 


-i 




20- 


J 




10- 


— 1 


o 
n 


10- 


^ 


2 




''::5 


H 
31 


20- 


-i 


n 




"::= 




30- 


-( 


O 




'?= 


39 


40-- 


-^ 



MOTION BY SLIDING CONTACT. 



121 



the number 49. The point {/, indicated on the drawing-board 
by the position of this number on the scale marked ''Scale of 
Centres for the Flanks of Teeth," is the centre required, from 
which the arc Tp must be drawn with the radius gT. The 
centre for the face Tn is found in a manner precisely simi- 
lar, by applying the slant edge of the instrument to the radial 
line AE. The number 21, obtained from the lower table, 
will indicate the position, /i, of the required centre on the 
lower scale. The arc Tn is then drawn, with h as a centre, 




Fig. so 



and radius TJi. "We have now the complete tooth outline for 
one side of one tooth ; the curve pTii being limited at the top 
by the addendum circle, and at the bottom by the root circle. 
Having proceeded thus far, the simplest way of drawing the 
rest of the tooth curves is to describe two circles about A, 
one through g and the other through h. Then all the centres 
for the flanks will lie on the former, and all the centres for 
the faces on the latter, of these two circles. We may now 
find these centres by striking from each pitch point an arc 
with radius equal io gT to cut the circle of centres for flanks, 
and an arc with radius Th to cut the circle of centres for 
faces. 

The curve nTp is also correct for an annular wheel of the 
same radius and number of teeth ; n becoming the root, and 



122 ELEMENTARY MECHANISM. 

p the point, of the tooth. Numbers for pitches not inserted 
in the table may be obtained by direct proportion from the 
column of some other pitch ; thus, for 4-inch pitch, by 
doubling those of 2-inch pitch. Also, no tabular numbers 
are given for 12 teeth in the upper table, because their flanks 
are radial lines. 

The variation in the contour, due to the addition of a 
single tooth, becomes less and less as the number of teeth 
increases ; so that the same curve will serve for wheels with 
nearly the same number of teeth. Consequently, if the num- 
ber assigned is not found in the tables, the nearest number 
found there is to be used instead. 

115.* Improved Willis Ocloiitograph. — In Fig. 80 
the points g^ /i, are found by drawing two radial lines, AD 
and AE^ and applying the instrument to each of them, or by 
drawing two additional lines, gD and Eli^ at an angle of 75° 
with AD and AE respectively, and setting off on them cer- 
tain lengths obtained from tables. Having found these 
points, circles of centres are drawn through them, and used 
as explained above. 

If, now, instead of proceeding in this manner, we could 
find from tables the radii of the two circles of centres, and the 
radii gT and T/i, the construction would be much simplified. 

This improvement is due to Mr. George B. Grant, who has 
calculated the distances of the two circles of centres from the 
pitch circle, and also the radii of the arcs for the faces and 
flanks. His results appear in the following table, where 
"Dis." represents the radial distance between the circle of 
centres and the pitch circle, and "Rad." the radius of the 
face or flank arc as the case may be : — 



* The tables in Arts. 115 and 116, and the substance of the matter in 
those articles, are taken, by permission, from " A Handbook on the 
Teeth of Gears," by George B. Grant, Boston, Mass. 



MOTION BY SLIDING CONTACT. 



123 



IMPROVED WILLIS ODONTOGRAPH TABLE. 

(Copyright, 1885, by George B. Grant.) 







For One 


Diametral 


For One-Inch Circular 






Pitch. 




Pitch. 


XuMBEK OF Teeth 
IN THE "Wheel. 










For any 


3ther Pitch, 


For any other Pitch, 






divide Tabular Value by 


multiply Tabular Value by 






that Pitch. 




that Pitch. 






Faces. 


Flanks. 


Faces. 


Flanks. 


Exact 


















Pad. 


Dis. 


Pad. 


Dis. 


Pad. 


Dis. 


Pad. 


Dis. 


12 


12 


2.30 


0.15 


_ 


_ 


0.73 


0.05 


_ 




13i 


13- 14 


2.35 


0.16 


15.42 


10.25 


0.75 


0.05 


4.92 


3.26 


15i 


15- 16 


2.40 


0.17 


8.38 


3.86 


0.77 


0.05 


2.66 


1.24 


m 


17- 18 


2.45 


0.18 


6.43 


2.35 


0.78 


0.06 


2.05 


0.75 


20 


19- 21 


2.50 


0.19 


5.38 


1.62 


0.80 


0.06 


1.72 


0.52 


23 


22- 24 


2.55 


0.21 


4.75 


1.23 


0.81 


0.07 


1.52 


0.39 


27 


25- 29 


2.61 


0.23 


4.31 


0.98 


0.83 


0.07 


1.36 


0.31 


33 


30- 36 


2.68 


0.25 


3.97 


0.79 


0.85 


0.08 


1.26 


0.26 


42 


37- 48 


2.75 


0.27 


3.69 


0,66 


0.88 


0.09 


1.18 


0.21 


58 


49- 72 


2.83 


0.30 


3.49 


0.57 


0.90 


0.10 


1.10 


0.18 


97 


73-144 


2.93 


0.33 


3.30 


0.49 


0.93 


0.11 


1.05 


0.15 


290 


145-rack 


3.04 


0.37 


3.18 


0.42 


0.97 


0.12 


1.01 


0.13 



This improved AYillis process will produce exactly the 
same circular arc as the usual method, with the same theo- 
retical error ; but its operation is simpler, and less liable to 
errors of manipulation. By this process the circles of cen- 
tres are drawn at once, without preliminary constructions, at 
the tabular distances from the pitch line ; and the table also 
gives the radii of the face and flank arcs. No special instru- 
ment is required, no angles or special lines are drawn to 
locate the centres,- and hence the chance of error is much 
less. 



124 ELEMEKTARY MECHANISM. 

116.* Grant's Odontograpli, — If, in the method de- 
scribed in the preceding article, we use, instead of the cir- 
cular arcs employed by Professor Willis, arcs which shall 
approximate still more closely to the true epicycloidal and 
hypocycloidal curves, we shall evidently obtain more satis- 
factory results. Mr. Grant has computed and tabulated the 
location of the centre of the circular arc that passes through 
the three most important points of the true curve; viz., at 
the pitch line, at the addendum line, and at a point midway 
between. The Willis arc runs altogether within the true 
curve, while the Grant arc crosses the curve twice. The 
average error of the Grant arc is much less than that of the 
Willis arc, and it is hence to be preferred. 

The circles of centres are drawn at the tabular distances, 
"Dis.," inside and outside the pitch line respectively; and 
all the faces and flanks are drawn from centres on these 
circles, with the dividers set to the tabular radii, "Rad.'* 
The tables are arranged in an equidistant series of twelve 
intervals. For ordinary purposes the tabular value of any 
interval can be used for any tooth in that interval ; but for 
greater precision it is exact only for the given "exact" 
number, and intermediate values must be taken for inter- 
mediate numbers of teeth. 

When the number of teeth is twelve, the flanks are radial, 
and hence no tabular values are given for the flanks of that 
number. 

To illus.trate the use of the following table, let it be re- 
quired to draw the tooth outline for a wheel of 24 teeth of 
IJ-inch pitch. Draw the pitch circle with its proper radius 
of 11.46 inches, and mark off the pitch points of the teeth. 
Draw the addendum, root, and clearance circles, having fixed 
on the dimensions of the tooth by means of some system of 
proportions such as those given in Art. 101. 

* See note on p. 122. 



MOTION BY SLIDING CONTACT. 



125 



GRANT'S ODONTOGRAPH TABLE. 

EPICYCLOIDAL TEETH. 
(Copyright, 1885, by George B. Grant.) 





1 


For One Diametral 


For One-Inch Circular 


Number of Teeth 
IN THE Wheel. 


Pitch. 




Pitch. 


For any 


Dther Pitch, 


For any other Pitch, 






divide Tabular Val 


Lie by 


multiply Tabular Value by 






that Pitch. 




that Pitch. 






Faces. 


Flanks. 


Faces. 


Flanks. 


Exact. 


Intervals. 












Rad. 


Dls. 


Rad. 


Dis. 


Rad. 


Dis. 


Rad. 


Dis. 


12 


12 


2.01 


0.06 


- 


_ 


0.64 


0.02 


_ 


_ 


m 


13- 14 


2.04 


0.07 


15.10 


9.43 


0.65 


0.02 


4.80 


3.00 


m 


15- 16 


2.10 


0.09 


7.86 


3.46 


0.67 


0.03 


2.50 


1.10 


m 


17- 18 


2.14 


0.11 


6.13 


2.20 


0.68 


0.04 


1.95 


0.70 


20 


19- 21 


2.20 


0.13 


5.12 


1.57 


0.70 


0.04 


1.63 


0.50 


23 


22- 24 


2.26 


0.15 


4.50 


1.13 


0.72 


0.05 


1.43 


0.36 


27 


25- 29 


2.33 


0.16 


4.10 


0.96 


0.74 


0.05 


1.30 


0.29 


33 


30- 36 


2.40 


0.19 


3.80 


0.72 


0.76 


0.06 


1.20 


0.23 


42 


37-48 


2.48 


0.22 


3.52 


0.63 


0.79 


0.07 


1.12 


0.20 


58 


49- 72 


2.60 


0.25 


3.33 


0.54 


0.83 


0.08 


1.06 


0.17 


97 


73-144 


2.83 


0.28 


3.14 


0.44 


0.90 


0.09 


1.00 


0.14 


290 


145-300 


2.92 


0.31 


3.00 


0.38 


0.93 


0.10 


0.95 


0.12 


GO 


Rack 


2.96 


0.34 


2.96 


0.34 


0.94 


0.11 


0.94 


0.11 



From the above table take the vahies given for the interval 
22-24 ; and, as the pitch is 1^ inches, multiply these tabular 
values by IJ. We then obtain 

Distance between pitch circle and circle of face centres = 0.07; 

face radius = 1.08. 
Distance between pitch circle and circle of flank centres = 0.54; 

flank radius =2.15. 

Draw the circle of face centres 0.07 inch inside the pitch 
circle, and the circle of flank centres 0.54 inch outside of 



126 ELEMENTARY MECHANISM. 

the pitch circle. With a pitch point as a centre, strike an 
arc with radius 1.08 inches to cut the circle of face centres, 
and an arc with radius 2.15 inches to cut the circle of flank 
centres. With these two points of intersection as centres, 
describe the face and flank through the pitch point, draw the 
same arcs in reversed position through a point on the pitch 
circle whose distance from the pitch point is the desired 
tooth thickness, connect the faces by an arc of the addendum 
circle, and join the flanks by fillets to the clearance circle, 
and the tooth is complete. 

This odontograph, as well as Willis', is arranged for an 
interchangeable set (Art. 98) , from a wheel with twelve teeth 
to a rack. 

117. Robinson's Templet Odontograpli. — In the use 
of this instrument, a method entirely different from those just 
mentioned is pursued. Instead of using circular arcs, the 
outlines of the teeth are drawn by means of a templet, which 
is the curved edge of the instrument itself, when the latter is 
brought into a proper position. 

As the epicycloidal curve is normal to the pitch line, and 
very nearly so to the .tangent to the pitch circle drawn from 
the middle of a tooth, it is clear that if a curve of rapidly 
changing curvature be so placed as to be normal to the tan- 
gent, as above described, and at the same time intersecting 
the addendum circle at the same point that the epicycloidal 
curve required for the tooth does, it will represent the epicy- 
cloidal tooth face with great precision. 

The curve adopted as conforming most closely, in general, 
with limited initial portions of the epicycloid, is the loga- 
rithmic sjnral. This curve appears to possess the highest 
degree of adaptation, because of its uniform rate of curvature, 
and also because this rate can be assumed at pleasure. In 
adopting the particular logarithmic spiral for the odontograph 
curve, inasmuch as this spiral may have an infinite variety 
of obliquities, it is evident that the selection is not a matter of 



MOTION BY SLIDING CONTACT. 127 

indifference. 'When the obliquity, or angle between the nor- 
mal and radius vector, is very small, the arc of this spiral 
changes curvature less rapidly than when the obliquity is 
great. When the obliquity is zero the spiral becomes a 
circle, and when it is 90° the spiral is simply a radius ; 
neither of which approximates to the desired curve. 

To find that obliquity which makes the spiral best fit the 
epicycloid, it will probably be most satisfactory to assume 
an epicycloid which represents an average of those likely to 
be used for both curves, and adapt the spiral to it, though 
any ordinary logarithmic spiral will evidently conform more 
closely to it than the circle. The spiral which most closely 
osculates the epicycloid for a pair of equal pitch circles is 
therefore adopted, because the opposite wheel may be either 
larger or smaller, thus making a higher or lower epicycloid. 

By an elaborate mathematical investigation,* Professor 
Eobinson has shown that this curve will produce the required 
results in all the various cases of epicycloidal and involute 
gearing. 

118. Manner of using" Odontograpli. — The instru- 
ment is shown in Fig. 81 of full size, and of suitable capacity 
for laying out all teeth below six inches pitch. The curved 
edge AB is the logarithmic spiral above spoken of ; and the 
curve AC is its evolute, in other words, an equal spiral. 

The instrument should be made of metal, because it is 
intended that it may be used directly for a scribe templet, in 
which use it will be subject to wear from the passes of the 
scribe. It has several holes in it, so that it may be attached 
by wood screws, or by bolts expressly prepared, to any con- 
venient wooden rod, in such a manner, that, when the rod 
swings around a centre-pin of the wheel, all the faces of the 
teeth may be described directly from the instrument itself. 



* For the complete mathematical discussion, see No. 24 Van Nos- 
trand's Science Series. 



128 ELEMENTARY MECHANISM. 




o 



u. 



ul 




o 



O 



o 



o 



o 



f 



MOTION BY SLIDING CONTACT. 129 

The desired result is thus obtained directly without the use 
of a pair of compasses. 

Accompanying the instrument are six different tables, 
varying according to the kind of tooth desired. One of the 
tables is for the teeth of wheels belonging to an interchange- 
able series ; the other tables are for variously curved flanks 
and for annular wheels. The manner of using all the tables 
is nearly the same, so it is simply necessary to indicate the 
method for any one of them. Fig. 82 shows the manner of 
using this odontograph to lay out the teeth of a wheel belong- 
ing to the interchangeable series. 

The table for this system is arranged in four columns, 
headed respectively, 1, "Diameter in Inches;" 2, "Num- 
ber of Teeth ; " 3, " Face Settings ; " 4, " Flank Settings." 
The two settings are given for one-inch pitch. 

In the figure, let MN be the pitch circle. If it is not 
given, it may be found by multiplying the pitch by the num- 
ber in the column "Diameter in Inches" corresponding to 
the number of teeth. 

Assume the point T as the middle of a tooth, and lay off 
TD = its half -thickness. At T draw the tangent tTf, and 
at D the tangent Dd. Make TH = TD. Take from the 
column ' ' Face Settings ' ' the figure corresponding to the 
number of teeth, and multiply it by the pitch ; this will give 
the setting number. Then place the graduated edge of the 
odontograph at H^ and in such position that the number and 
division of the scale shall come precisely on the tangent line 
at i7, while at the same time the other curved edge is tangent 
to the line tTt\ The tooth outline is then traced along 
the instrument from D as far as needed. By turning over the 
instrument, which is graduated on both sides, and repeating 
the operation, we get the opposite face of the same tooth. 

To draw the flank, find a similar setting number by using 
the column "Flank Settings." The instrument is to be set 
with the division at i), and the other curved edge tangent to 



130 ELEMENTARY MECHANISM. 

Dd \ and the flank may then be drawn to the proper depth. 
When it is desired to repeat the operation of drawing the 
curves all around the wheel, the simplest way to locate the 
instrument is by drawing circles through the points A and C 
when it is once properly located. The instrument can then 
be readily placed at any tooth outline by placing the gradu- 
ated edge on the pitch point, and keeping the points A and C 
in the circles just mentioned. 

For instance, let it be required to draw the teeth of a wheel 
having 50 teeth of 3-inch pitch. For this number of teeth 
we find the tabular values : — 



Diam. in Inches. No. of Teeth. Face Setting. Flank Setting. 

15.917 50 0.42 0.66 



The diameter of the pitch circle is 3 x 15.917 = 47.751 = 
47f inches. The proper setting to draw the face is 3 x 0.42 
= 1.26, and the corresponding setting for the flank is 3 x 
0.66 = 1.98. 

Hence, to draw the face, the odontograph is placed so that 
the number 1.26 on the scale is at the point H (Fig. 82) ; 
and, to draw the flanks, it is placed so as to bring the number 
1.98 at D, 



MOTION BY SLIDING CONTACT. 131 



CHAPTER VIII. 

COMMUNICATION OF MOTION BY SLIDING CONTACT. 

VELOCITY RATIO CONSTANT. 

DIRECTIONAL RELATION CONSTANT. 

TEETH OF WHEELS (CONCLUDED). 

Pin Gearing. — Low-Numhered Pinions. — Unsymmetrical Teeth. — 
Twisted Gearing. — Non-Circular Wheels. — Bevel Gearing. — 
Skew-Bevel Gearing. — Face Gearing. 

119. Pin Gearing. — In Art. 76 it has been shown 
that an epicycloid traced on the pitch circle of the driver, by 
rolling on the latter a describing circle equal to the pitch 
circle of the follower, will drive a pin in the circumference 
of the following pitch circle with the same constant velocity 
ratio as if the pitch circles rolled together. 

In Fig. 83, let MN and ES be the pitch circles. Lay off 
the equal pitch arcs Ta and Tb ; and, with ItS as the 
describing circle, trace through a the epicycloid aD, which 
will, of course, pass through 5. Draw the equal epicycloid 
Tl) in reverse position through T, and let D be the point of 
intersection of the two epicycloids. Then aDT is the com- 
plete outline of a tooth of MJSf which will drive a pin b 
(having no appreciable diameter) on ES with the constant 

a A.T 
velocity ratio — = - — . Through D draw the arc DP con- 

centric with MN. The point P, where this arc intersects 
BS.) will evidently be the point at which the tooth aDT and 



132 



ELEMENTARY MECHANISM. 



the pin h will quit contact. Through P draw the epicycloid 
Pa' equal to Da\ then TP = Ta' = arc of recess. The 
wheel MN moving as indicated by the arrow, the contact 
will begin at T, and the point of contact will travel to the 
right, along the arc TP, until it reaches the point P, where 
contact ceases. The contact is wholly on one side of the 
line of centres ; and when the teeth drive^ as they should 
always do (Art. 90), there is no arc of approach. 




120. With given pitch circles, to find the relation between 
the arc of recess and the pitch. 

In Fig. 83, let TP^ the arc of recess, be given. Through 
P describe the epicycloid Pa' by rolling US on MN; draw 
the radius PA^ intersecting MN in K. Then, in order to 
secure the desired arc of recess^ the pitch must not be greater 
than Ta' = TP, nor less than 2Ka'. If Ta {= 2Ka') be the 
pitch, and the tooth be pointed, the arc of recess will be 



MOTION BY SLIDING CONTACT. 133 

jPP, as required. If, with the same pitch, the tooth be given 
some thickness at the top, the arc of recess will become less ; 
and, when the latter has its smallest value (i.e., when it is 
just equal to the pitch), the top of the tooth will be cut off 
so as to give the tooth outline Tcba. 

121. If the pitch be given, and it is required to find the 
arc of action which may be secured, lay off on MN the given 
pitch arc Ta, and, with RS as describing circle, construct 
the epicycloids TD and aD. Through their point of inter- 
section, D^ draw the arc DP concentric with MN. Then TP 
is the maximum, and Tb {= pitch) is the minimum, value 
of the arc of recess ; the tooth in the former case being 
pointed, and in the latter cut off at cb. 

122, Pins of Sensible Diameter. — In the preceding 
articles we have treated the pins as mere mathematical lines ; 
but in practice they must, of course, be given some magni- 
tude, and they are usually made as cylinders of a diameter 
of about half the pitch. The form of the tootJi must then 
be so modified, that, when it acts on the cylindrical surface of 
the pin, the latter shall move just as though its axis were 
being driven by the original epicycloid ; in other words, the 
constant normal distance from the latter to the new tooth 
outline must be equal to the radius of the pin. The manner 
of finding this derived curve is shown in Fig. 83. About 
successive points along the epicycloid, as centres, circular 
arcs are drawn, having the same radius as the pin ; a curve 
drawn tangent to this series of arcs will be the required tooth 
outline. 

In deriving the new tooth outline to act with a pin of sen- 
sible diameter, the length of the driver's tooth has evidently 
been reduced, causing a certain diminution of the arc of 
recess. Now, assuming the derived curve to be an epicy- 
cloid, identical with the original epicycloid, but simply moved 
in position, it is evident that contact will begin just as the 
centre of the pin reaches the point T; in other words, an 



134 ELEMENTARY MECHANISM. 

arc of approach will have been introduced practically equal 
to the radius of the pin. Now, although this assumption as 
to the shape of the derived curve is not strictly true, yet the 
error thereby introduced is inappreciable, and of no impor- 
tance in any practical case. 

123. Limiting- Diameter of Pin. — In the practical 
construction of problems in pin gearing, it sometimes becornes 
important to determine the maximum diameter of pin that 
can be used under given conditions. For instance, let the 
diameters of the pitch circles and the pitch be given, and let 
it be required to determine the maximum diameter of pin 
which will secure a certain arc of action. In Fig. 84, let 
MN and RS be the given pitch circles, and let the given 
pitch be f inch. Let the required arc of action be 1 J times 
the pitch ; that is, 1|- x f = xf inch. Lay off the arcs Ta 
and Tb^ each equal to the desired arc of action, and let abe 
be the epicycloid which would be described by the point h in 
rolling the circle RS on the outside of MN. Join Tb by a 
straight line, which will be normal to the epicycloid abe at 
the point b. Now, as the contact between the derived tooth 
and the cylindrical pin begius when the centre of the latter is 
at r, the desired arc of action w^ill be secured if contact 
ceases when the centre of the pin reaches b. Hence 6 will 
be the position of the centre of the pin at the moment of quit- 
ting contact, and the point of the tooth must evidently lie on 
the line Tb at a distance from b equal to the radius of the 
pin. 

But as «c is the pitch arc, the point of the tooth must evi- 
dently also lie on a radius bisecting this pitch arc ; and it 
will consequently be found at P, the intersection of these 
two lines. Pb then will be the radius that will secure the 
given arc of action on the supposition that the teeth are 
pointed. 

But if, as is usually the case, it is desired that the teeth 
should have some thickness at the top, the radius of the pin 



MOTION BY SLIDING CONTACT. 



135 



mast be made somewhat smaller than the maximum radius 
just found. 

Thus, in the present example, let us determine the shape 
of the tooth which will give the same arc of action with a 
smaller pin. Let the pin be made of the usual diameter 
employed in practice ; namely, i pitch. 




In Fig. 84, on the left of the centre line AB, lay off Ta' = 
Th' = arc of action, and aV = pitch, as before. About h' 
as a centre, describe the circle of the pin with a radius, h'P', 



136 ELEMENTARY MECHANISM. 

of one-fourth the pitch. Now, although P' will evidently be 
the extremity of the derived tooth curve, yet it will be found 
that the radius passing through that point will no longer bi- 
sect aV. In fact, the tooth now has some thickness at the 
top ; and drawing, in a reversed position on the other side of 
the middle point of aV, the derived curve found for this 
diameter of pin, and joining the two curves at the top by a 
circular arc through P' and concentric with MN^ we have the 
complete tooth outline for this case. 

124. Practical Example. — In Fig. 85 is shown a prac- 
tical example of pin gearing, the diagram being drawn full 
size, and being the solution of the following problem : — 

Distance between centres of pitch circles, 9 inches. Driver 
to have 50 teeth ; follower 40 pins. Arc of action to be 2 J 
times the pitch. Dividing the line of centres at T in accord- 
ance with the given number of teeth, we find the radius of 
the driver to be 5 inches, and that of the follower to be 4 
inches. 

Rolling the circle RS as a describing circle on the outside 
of MN^ the point T will describe the epicycloid Tt^ which 
which will be the form of a tooth of MN that would work 
with a pin of no appreciable diameter on RS. To find the 
maximum diameter of pin that will secure the desired arc of 
action, we proceed as in Art. 123. Lay off the arcs Ta and 
Th^ each equal to the required arc of action, i.e., 2 J times 
the pitch ; and lay off aa' equal to the pitch. Joining T6, 
and drawing the radius bisecting aa\ we find the maximum 
size of the pin and the corresponding shape of the tooth, as 
shown in dotted lines. We may, of course, use any radius 
of pin less than P&, and still secure the desired arc of action. 
In order to get teeth of better proportions, let us make the 
radius of the pin equal to one-fourth the pitch. On making 
a construction similar to the one explained in the latter part 
of Art. 123, we will find the shape of the tooth as finally 
drawn in the diagram. Just as in other gearing, clearance 



MOTION BY SLIDING CONTACT. 



137 



must be given at the bottom of the spaces, and this is usually 
done by means of circular arcs, as shown. 

The principal advantages of pin gearing are its smooth- 
ness of action, and the facility with which the pins may be 
turned in a lathe. 




Fig:. 85 



The pin wheel is often made of two plates, the ends of the 
pins being fixed into equi-distant holes in both plates, thus 
making a very strong arrangement, and one w^hich is fre- 
quently employed in clock-work. Such wheels are called 
lanterns or trundles, and their pins are called staves. 

125. Rack and Wheel. — -As previously stated, the 
2nns are alwaj^s given to the follower, and hence this com- 
bination will present two cases according to whether the 
rack is driver or follower. In Fig. 86 the rack drives and 
the wheel carries the pins. The teeth of the rack are formed 



138 ELEMENTARY MECHANISM. 

by curves parallel to the cycloids which would work correctly 
with the axes of the pins. In Fig. 87 the wheel drives and 




:Fig. 8Q 

Ihe rack carries the pins. The teeth of the wheel are formed 
by curves parallel to the involutes of its own pitch circle, 
which would work correctly with the axes of the pins. 




126. Annular Wheels. — If the annular wheel drives, 
as in Fig. 88, the pins are given to the small wheel, and the 
teeth of the annular wheel are formed by curves parallel to 
the hypocycloids which would work correctly with the axes 
of the pins. If the annular wheel is the follower, as in 
Fig. 89, it carries the pins ; and the teeth of the small wheel 
are formed by curves parallel to the epicycloids which would 
work correctly with the axes of the pins. 

When the annular wheel is the driver, and is twice as large 
as the wheel with which it gears, the h3q)ocycloids become 
straight lines, and the parallel tooth outlines will evidently 
also be straight lines. 



MOTION BY SLIDING CONTACT. 



139 



Fig. 90 shows such tin arrangement, in which the pin 
wheel has but three pins, while the wheel teeth are formed 




T^ig. 88 



by cutting three straight grooves, intersecting each other at 
the centre of the wheel, at angles of sixty degrees, each 




-Fig. 89 



being of a width equal to the diameter of a pin. By placing 
rollers on the pins, and making the widths of the slots equal 



Fig. 90 




to the diameter of these rollers, this arrangement of pin 



140 ELEMENTAKY MECHANISM. 

gearing can be used as a shaft eonpling to drive in either 
direction. 

127. Low-Numbered Pinions. — As the number of 
teeth in a wlieel decreases, the teeth themselves becom.e 
longer, and both the obliquit}^ of action and tlie amount of 
sliding rapidly increase. Pinions having very few teeth are, 
for these reasons, unsuitable for general use ; and accord- 
ingly we find that in practice no wheel of less than about 
twelve teeth is employed if it can possibly l)e avoided. In 
order to secure smoothness of action and a minimum obliq- 
uity of pressure, the number of teeth assigned to any given 
wheel is usually so great that no doubt exists as to their 
successful working. It occasionally happens, however, that 
it becomes imperatively necessary to employ wheels having 
as few teeth as possible ; and it then becomes a matter 
of importance to determine whether the desired numbers of 
teeth will work together. 

128. Practical Example. — The practicability of any 
assumed case can be readily determined by the construction 
of a diagram, keeping in mind the limitations as to pitch, arc 
of action, etc., explained in previous articles. For example, 
let us examine the case of two pinions, of five and seven 
teeth respectively, and liaving radial flanks. 

In Fig. 91, let A and i> be the centres of the pitch 
circles, and T their point of tangency. As the flanks are 
to be radial, the diameters of tlie describing circles will be 
equal to the radii of the respective pitch circles, as sliown. 
Assume the arc of action to have its smallest value (namely, 
just equal to the pitch arc) , and let the arcs of approach and 
recess be equal. Constructing the teeth under these condi- 
tions (Art. 97), we will obtain the wheels shown in Fig. 91. 
These wheels will just barely work, one pair of teeth quitting 
contact at P at the same instant that another pair are coming 
into contact at F\ It is evident that, by continuing the 
opposite faces until they meet, the arc of action can be some- 



MOTION BY SLIDING CONTACT. 



141 




Fig. 91 



what increased without making any other change. Two such 
wheels, then, can be made to work, though they will never 
run with the smoothness of action that characterizes wheels 
having a large number of teeth. 



142 ELEMENTARY MECHANISM. 

129. In Fig. 91, let PT represent in magnitude, as it 
does in direction, the pressure between a pair of teeth at the 
moment of quitting contact, the angle PTf being thirty-six 
degrees. Then Ph evidently represents the component of 
PT which tends to force the axes apart, thus producing fric- 
tion and wear in the bearings. Now% callings the tangential 
pressure necessary to transmit any given power, it is evident 

that in this case PT = p x ^= 1.24 p, and Ph = PT sin 

36° = .73 p. The obliquity has thus caused a pressure be- 
tween the axes almost three-fourths as great as the pressure 
producing rotation. It is evident that there is a limit beyond 
which the angle of maximum obliquity cannot go without 
increasing the prejudicial component Ph to an inordinate ex- 
tent. The limit for the mean obliquity is usually placed at 
fifteen, and that for the maximum, at thirty degrees ; though, 
where the pressure to be transmitted is not great, thirty-six 
degrees may be made tlie limit for the latter. In the case 
of involute teeth, the obliquity is constant, and should never 
exceed fifteen degrees. 

The maximum diameter of the describing circle (Art. 94) 
is usually takeii as half the diameter of the pitch circle in 
which it rolls ; but, for special reasons, it may be increased 
to five-eighths of that diameter. By repeating the construc- 
tion within these limits of obliquity and of size of describing 
circle, we will find that Jive is the least number of teeth for 
each of two equal pinions, that four will work with Jive or 
any greater number, that three will work with any number 
greater than fourteen, and that less than three cannot be 
made to work at all. 

130. Two-Leaved Pinion There is, however, an 

exception to the last statement ; for if the teeth are placed 
in parallel planes, instead of, as usual, in the same plane, a 
two-leaved pinion can be made to drive in a very satisfactory 
manner. 



MOTION BY SLIDING CONTACT. 143 

This arrangement is shown in Fig. 92. B represents a 
disc, to which teeth 6, h\ 6, h\ etc., are fixed alternately on 
opposite sides. The acting surfaces of these teeth are 
straight, and radiate in direction from the centre of B. The 
driver is formed of a pair of double epicycloids, of which 
A is in the plane of the teeth 5, 6, etc., and A' is in the 
plane of the teeth h'^ b% etc. The radius of the pitch 
circle of B, and hence the diameter of the describing circle 
for the teeth, is equal to the distance from the centre of B 
to the outer extremity of one of its teeth. 




Fig* 93 

The action is not very oblique, but the amount of sliding 
is considerable. As the driver has onl}^ faces, and the fol- 
lower only flanks, the action takes place on one side of the 
line of centres only. The combination is always used so 
that the action may be receding ; and the result is, that the 
motion is exceptionally smooth and noiseless. 

A pinion of one tooth communicating a constant angular 
velocit}^ ratio between parallel axes, appears absolutely im- 
possible. The endless screw is equivalent, however, to a 
pinion of a single tooth. 



144 



ELEMENTARY MECHANISM. 



An arrangement similar to tliat of Fig. 92 may also be 
employed to give motion to a rack, as shown in Fig. 93. 
The construction in this case is simplified, the curves of the 
pinion becoming involutes of the pitch circle, while the rack 
consists of a straight bar, having equal rectangular pieces 
fixed to it at regular intervals on both sides. 




r'ig. 93 



This arrangement is objectionable on account of the wear 
being confined to a single point, just as was explained in the 
case of Fig. 70. Hence, in any practical case it would be 
better to use a describing circle of comparatively small di- 
ameter, making the outline of the rack teeth cycloidal in 
shape, and thus distributing the action over a greater amount 
of surface. 

131. In Arts. 128 and 129 we have assumed the arcs of 
approach and of recess to be equal ; that is, each equal to 
half the pitch. Though the total arc of action cannot be 
less than the pitch, yet we may evidently vary the relative 
amounts of approaching and receding action at pleasure. 
If, as is usually the case, the arc of recess is to be the 
greater, it is evident that a pinion of fewer teeth can be 
used to drive than to follow ; for the arc of recess depends 
upon the length of the driver's teeth, and this length again 
depends on the size of the describing circle. If we take a 
given wheel and pinion, and gradually decrease the number 
of teeth in the pinion, — in other words, make it of a smaller 



MOTIOX BY SLIDING CONTACT. 145 

diameter, — we shall evidently also decrease the diameter of 
the describing circle, which g-enerates the faces of the wheel 
teeth, and hence diminish their lengths. The generating circle 
for the faces of the pinion's teeth will not l)e affected, so that 
the length of these teeth will remain almost the same. 

But, whatever the conditions, we can evidently determine 
the practicability of any given case by the construction of a 
diagram, as explained. 

132. Least Follower for g-iven Driver. — Though the 
construction of a diagram will always enable us to ascertain 
whether a given combination of driver and follower is pos- 
sible, yet the problem may sometimes be stated in a more 
general form. For example, let it be required to find the 
least number of teeth that can be given to a wheel wiiich is 
to follow a given driver. Let the pitch circle, pitch, and arc 
of recess of the driver be given as in Fig. 65. Taking the 
upper describing circle, as given in the diagram, it is evident 
that the follower may have less teeth than there shown. For 
we may make the radius of the follower's pitch circle only 
twice TC^ in which case the follower's teeth will have radial 
flanks. 

Now, if the driver's teeth were to be pointed, instead of 
as shown in the diagram, it is clear that with the given pitch, 
the required arc of recess could be secured with a smaller 
describing circle ; and hence a smaller follower could be 
used. In this case, at the moment of quitting contact, the 
point of the driver's tooth must lie at the intersection of the 
upper describing circle with a radius of MN bisecting Ha., 
and at a distance from T, measured on the circumference of 
the describing circle, equal to the arc of recess. Hence, 
drawing the radius bisecting Ha^ and finding (Art. 81) the 
position of this point, we describe through the latter and 
the point T a circle whose centre lies on AB. This will be 
the upper describing circle required. If we then make the 
diameter of the follower's pitch circle twice the diameter of 



146 ELEMENTARY MECHANISM. 

the describing circle thus found, we shall evidently have 
determined the required least number of teeth, on the assump- 
tion that they are to have radial flanks. As previously 
mentioned, the diameter of the pitch circle may, for special 
reasons, be made as small as | of that of the describing 
circle ; but the size of this pitch circle must always be such 
that the given pitch will be an aliquot i)art of the circumfer- 
ence. Having thus determined the least number of teeth, 
we must ascertain if the obliquity is within the desired 
limits. If it is found to exceed the assigned limit, the size 
of the pitch and describing circles must be increased until 
the obliquity is reduced to the proper amount. 

133. Again, the assigned conditions may belong to the 
follower, and it may be required to determine the least num- 
ber of teeth for the driver. This case may be solved by an 
obvious modification of the above process. Both of these 
cases will also present themselves in the involute sj^stem. 
With involute teeth, the maximum arc of recess will evi- 
dently be secured if the driver's tooth be pointed, and if 
its point touch the base of the follower's involute at the 
moment of quitting contact. The obliquity of action which 
secures this result is the greatest possible, and gives the 
minimum foUoiver that can be employed. 

Similar problems will occur in regard to annular wheels ; 
and such problems may be solved by similar methods, the 
only peculiarity being, that there will be both maximum and 
minimum values. 

The least annular wheel which can be driven by a given 
pinion must have one tooth more than the pinion. The 
smallest pinion which can be thus used is one of three teeth, 
the wheel then having four teeth. 

The least annular wheel that can drive a given pinion must 
have one and a half times as many teeth as the pinion when 
the latter has radial flanks. The various questions of limit- 
ing numbers may also be solved in pin gearing, though the 



Motion by sliding contact. 



14T 



method is more complex, owing to the pecuUar nature of the 
derived cnrve. 

134. But, in any case, it is simply a question of graphical 
construction ; the teeth being laid out in accordance with the 
prescribed conditions. Tables have been prepared giving 
such least numbers, calculated with considerable exactness, 
for various arcs of recess ; and though it is always prefer- 
al)le to make the graphic construction for the special case 
under consideration, yet the following brief extract from 
such tables may not l)e without interest. In these tables 
the flanks of all the spur wlieels are supposed to be radial, 
and the thickness of the tooth and the width of the space, 
measured along the pitch circle, are supposed to be equal. 



EPICYCLOIDAL GEARING. 

TABLE OF THE LEAST-NL^IBERED SPUR \VHEELS AND GREATEST- 
NUilBERED ANNULAR WHEELS THAT WILL WORK WITH GIVEN 
PINIONS. 







Least Number of Teeth in 


Greatest Number of 




Number of 


Spur Wheel 


Teeth in Annular Wheel 




Teeth ia 
Given 


. 
















Pinion. 


If Wheel 


If Pinion 


If Wheel 


If Pinion 






Drives. 


Drives. 


Drives. 


Drives. 


^ 


2 


impossible 


impossible 


impossible 


7 


^ 


.3 


i( 


a 


'^ 


41 




4 


a 


34 


a 


rack 


II 


5 


a 


19 


12 




ai 


6 


" 


14 


65 




O 


7 


33 


1.2 


rack 




V. 


8 


16 


11 






p 


9 


11 


10 






< 


10 


9 


10 







148 ELEMENTARY MECHANISM. 

EPICYCLOIDAL GEARING (Concluded). 







Least Number of Teeth in 


Greatest Number of 




Number of 

Teeth in 

Given 

Pinion. 


Spur Wheel 


Teeth in Annular Wheel 


If Wheel 


If Pinion 


If Wheel 


If Pinion 






Drives. 


Drives. 


Drives. 


Drives. 


;d 












'a 


2 
3 


impossible 


impossible 
37 


impossible 


11 

rack 


II 


4 


u 


15 


8 




m 


5 


a 


11 


53 




9 


6 


21 


10 


rack 




1 


i 


11 


9 






8 


8 


8 







135. Unsym metrical Teeth. — In all the figures of 
teeth hitherto given, the teeth are symmetrical, so that they 
will act whether the wheels be turned one way or the other. 
If the machine be of such a nature that the wheels are to be 
required to turn in one direction only, the strength of the 
teeth may be greatly increased by an alteration iii form first 
suggested by Professor Willis. In Fig. 94 are represented 
two wheels, of which the lower is the driver, and always 
moves in the direction of the arrow. The describing circles 
are made large, thus reducing the obliquity of action. The 
right side of the driver's teeth and the left side of the fol- 
lower's teeth are the ouly portions that are ever called into 
action ; and they are made precisely as usual in the epicy- 
cloidal system. If the other sides were made the same, this 
would give a very weak form at the root. To obviate this, 
the back of each tooth is bounded by an arc of an involute. 
The bases of these involutes being proportional to the pitch 
circles, they will during the motion be sure to clear each 
other, because, geometrically speaking, they would, if the 



MOTION BY SLIDING CONTACT. 149 

wheels moved the other way, work together correctly, though 
the inclination of their common normal to the line of centres 
is too great for the transmission of pressure. The effect of 
this shape is to produce a very strong form of tooth l^}' 
taking away matter from the extremity of the tooth where 
the ordinary form has more than is required for strength, 
and addins; it to the root. 




Fig. Q4= 

136. Twisted GeariDg' In this class of gearing (Art. 

58) the point of contact travels, during the motion of the 
wheels, from one side to the other. The outer planes of 
the wheel should be twisted through an angle equal to the 
pitch, so that a fresh contact is always beginning on one side 
as 1-he last contact is quitting on the other. In the double 
wheel shown in Fig. 39, there are, of course, two points of 
contact, travelling in a S3nnmetrical manner with respect to 
the mid-plane of the wheel. The teeth must be so formed, 
that, when the angular velocity ratio is constant, contact shall 
only take place at the instant of crossing the line of centres. 
Otherwise, if the teeth were formed upon the usual princi- 



150 ELEMENTARY MECHANISM. 

pies, it is evident that the sliding contact before and after 
the line of centres would still remain. This may evidently 
be accomplished by making the flanks by any of the usual 
methods, and then making the faces so that they will lie 
icithiu the faces which would be proper for a spur wheel with 
the flanks assumed. The simplest mode of making such 
teeth is to give them radial flanks, and make the faces semi- 
circles whose diameter is the thickness of the tooth at the 
pitch circle. The motion is now transmitted by pure rolling 
contact, and the action of these wheels is exceedingly smooth 
and noiseless. They are, however, better suited for light 
work, because the pressure is confined to a single point, 
instead of being distributed along a line. For heavy work 
it is preferable to employ the stepped wheels (Fig. 37) in 
which the teeth are of the usual forms for spur wheels. In 
this case, the motion is, of course, no longer transmitted by 
pure rolling contact ; but the action is, nevertheless, much 
smoother than that of ordinary spur wheels. 

137. ]S"on-circular Wheels. — In all the preceding cases 
of toothed wheels the pitch curves of the wheels have been 
circles ; but the teeth may be just as well laid out when the 
pitch curves are not circular, though in the latter case the 
operation is much more tedious. 

The two pitch curves must, in any case, be capable of 
rolling together with a constant velocity ratio. For instance, 
let it be required to lay out the teeth of a pair of equal 
ellipses. Divide the perimeter of the ellipse for the location 
of the teeth and the spaces. Find, by trial and error, the 
centre of curvature of the ellipse at the point where it is 
desired to draw a tooth outline. The tooth outline may then 
be drawn by rolling within and without the pitch ellipse a 
describing circle in the usual manner ; the actual operation 
being performed by substituting for the pitch ellipse a circle 
whose radius is the radius of curvature of the ellipse at the 
point considered. By repeating this operation at successive 



MOTION BY SLIDING CONTACT. 151 

l)iteli points, we can thus draw all the teeth. This method 
is perfectly general, and may be applied to rolling curves of 
any form, such as, for instance, the lobed wheels shown 
in Figs. 43 to 46. If the same describing circle be used 
throughout, its diameter should be such as to give radial 
flanks to the teeth in that part of the pitch line where the 
curvature is sharpest. Should other parts be very much 
flatter, the flanks of the teeth may spread too rapidly. This 
may be remedied b}^ using different describing circles for 
the teeth in those parts, care being taken that the same one 
be always used for the face and flank that are to work 
together. 

If one of the wheels be made a pin wheel, its pitch curve 
is to be used as the describing curve to generate the teeth of 
the other. 

138. Bevel Wheels. — In all the cases of wheels pre- 
viously considered, the pitch surfaces have been cylinders, 
all the transverse sections being consequently alike. Hence 
it was found most convenient to deal with one such section, 
so that the problems involved only lines instead of surfaces. 
But the pitch and describing curves employed, as well as the 
tooth outlines constructed, are merely transverse sections of 
surfaces whose elements are parallel to the axis of the wheel. 
Considering the cylinder as the special case of the cone in 
which the vertex is removed to an infinite distance, it would 
seem, that, in the case of the cone, the elements of the 
analogous surfaces should converge to the vertex of the cone. 

In other words, just as we roll a describing cylinder within 
and without a pitch cylinder to generate the tooth surfaces of 
spur wheels, so may we roll a describing cone within and 
without 3i pitch cone to generate the tooth surfaces of bevel 
wheels. In both cases the line of contact of the tooth sur- 
faces will be a right line ; in the former it will be parallel to 
the axis of the cylinder, and in the latter it will pass through 
the vertex of the cone. 



152 



ELEMENTARY MECHANISM. 



139. In Fig. 95, let CDTE be the pitch cone, and CPTH 
the describing cone ; the two cones having the common 
vertex C, and being in contact along the right line CT. 
Draw any element, such as (7P, of the describing cone, and 
consider the latter to roll to the left, keeping its vertex at (7, 
and remaining always in contact with the pitch cone. CT is 
at any moment the instantaneous axis about which the plane 
CPT revolves ; hence the surface (7Pa, generated by the 
line OP, will be normal to the plane OPT. 




We have seen that, with parallel axe«, \Aoc\x ^\ii-vt,9> may 
be selected which will produce a variable velocity ratio. 
Similarly, in bevel wheels, the bases of tlie cones might be 
so shaped as to produce changes iii the velocity ratio. In 
practice, however, this is never done ; any desired variation 
of velocity ratio being produce(? by some other means. We 
may, therefore, confine our attention to the case in which all 
the cones have circular btises. In this case, the point P, 
being at a constant distaiice from O, will move in the surface 
of a sphere of which is the centre, and whose radius is 
egual to the slant height of the cones. The arc TP — arc 



MOTION BY SLIDING CONTACT. 153 

Ta ; and the curve Pa^ descril)ed li}^ the point P, is a spher- 
ical epicycloid. Similarly, b}^ rolling the describing cone 
within a pitch cone, a spherical hypocycloid will be generated. 
F'ollowing out the analogy between cylinder and cones, it is 
evident that, just as the tooth surfaces of cylindrical wheels 
are formed by moving a*right line along the epicycloid and 
hypocycloid previously discussed, keeping the line always 
parallel to the axis of the pitch cylinder, so the tooth sur- 
faces of conical wheels must be formed by moving a right 
line along the spherical epicycloid and hypocycloid, making 
the line in this case always pass through the common vertex 
of the pitch cones. 

140. Construction of Tooth Outline. — The portion 
of the spherical surface occupied by the spherical epic^^cloid 
and hypocycloid, when they are used in the formation of 
teeth, is a narrow zone extending a short distance on both 
sides of the base circle of the pitch cone. For all practical 
purposes we may substitute for this narrow spherical zone a 
portion of the surface of a cone which is tangent to the 
sphere in the base circle of the pitch cone, and whose ele- 
ments are consequently perpendicular to the corresponding 
elements of the pitch cone. 

In Fig. 96, let CA and CB be the given axes of the pitch 
cones. Dividing the angle ACB so as to obtain the required 

r q 

A^elocity ratio, which in this case is - = -, we find CP, the 

a 2 

common element. The bases FGP and EHP are evidently 
small circles of the sphere whose radius is CP. Draw PA 
perpendicular to (7P, and revolve it around the axis CA, 
generating the normal cone FPA. Similarly, draw PB 
perpendicular to OP, and revolve it about the axis CB., 
generating the normal cone PBE. 

These new cones comply with the conditions above men- 
tioned, and a narrow zone of their curved surfaces may be 
used upon which to describe the tooth outlines. 



154 



ELEMENTARY MECHANISM. 



If, now, we roll a describing cone without one of the pitch 
cones and within the other, we will generate the tooth sur- 
face for the faces of the former and for the flanks of the 
latter. In order to construct this surface, we must select 
some particular element of the describing cone, and find the 
curve which it describes on the surfaces of the normal cones. 
To do this, we need only draw this element in successive 
positions, and find the points in which it pierces the normal 
cones. The curve formed by joining these successive points 
will be the directrix of the tooth surface ; and the latter will 
be formed by moving a straight line along this generatrix, 
the line always passing through the common vertex of the 
pitch cones. 




Fi^. QG 



This method will give the exact curves ; the error of using 
the surface of the normal coue, instead of that of the sphere, 
being so small as to be inappreciable. Its application to 
practical cases involves more labor, however, than that of 
the following approximate method, which is the one in 
almost universal use. 

141. Tredg-old's Method. — If we assume the curved 
surface of each of the normal cones to be cut along one of 
the elements, and spread out on a plane, we will have (Fig. 
96) portions of two circles whose radii, A'F^ and B'P% are 



MOTION BY SLIDING CONTACT. 



155 



the slant heights of the cones. If, now, these circles be 
taken as pitch circles, and teeth be constructed on them by 
any of the usual methods for spur wheels, we may then wrap 
these surfaces, with the teeth, back into their original conical 
shape ; and using the tooth curves, as they then appear on 
the normal cones, as directrices, we may generate the re- 
quired tooth surfaces by moving a right line in contact with 
the curves, and passing through the common vertex of the 
cones, as before. 

142. The practical method of drawing such teeth is shown 
in Fig. 97. Let AC be the axis of the bevel wheel, let CDE 
be the pitch cone, and AED the normal cone ; DJSfE being 
the circular base common to both cones. 




In the side view, draw a line parallel to AD^ and project 
the latter on it at A'D\ With centre A^ and radius A'D' ^ 
describe a circular arc which will be an arc of the pitch circle 
to be used. On this arc lay off a tooth by the usual method, 
being careful to make the pitch an aliquot part of the cir- 
cumference of a circle whose radius is ND. The tooth 



156 ELEMENTARY MECHANISM. 

outlines may then be drawn by means of describing circles or 
by the approximate odontograph methods, according to the 
degree of accuracy required. Project A'K' ^ the radius of 
the root circle, at AK^ and AH\ the radius of the addendum 
circle, at AH. The points iJ, D, /i, of the line AH will, 
by revolution, describe circles about AC^ which will be repre- 
sented in the side view by the straight lines FH^ ED, and 
LK, and which will be seen in their true size and shape in 
the end view. 

On the end view of the circles just mentioned we must next 
lay off, on each side of a radius, the half-thickness of the 
tooth at the top, at the pitch line, and at the bottom, as 
obtained from the development. If great accuracy is re- 
quired, any number of additional circles may be used in a 
similar manner. Having thus determined the end view of 
the tooth outlines, we must next project each one to tlie side 
view ; the points lying in each circle being projected to the 
straight line which is the side view of that circle. In prac- 
tice, only frusta of conical wheels are employed, and the 
teeth are limited at both ends by normal cones. It is evi- 
dent that in this case the shape of the teeth will be similar 
at both ends, except that the outer ones will be larger in 
proportion to their greater distance from the vertex. The 
points of the inner tooth outlines are found by drawing radii 
through the principal points of the outer tooth outlines already 
determined, and finding the intersection of these radii with 
the circles corresponding to the inner normal cone. 

It may be required to describe the teeth by either the epi- 
cycloidal or the involute system, or so that they may be used 
for an annular bevel wheel ; but the modification of the 
general operation is in each case similar to the correspond- 
ing modification for wheels on parallel axes. 

143. RelatiA e Action of Bevel and Spur Wheels. — 
The action of a toothed wheel, other things being equal, is 
always more smooth iu proportion as the teeth increase in 



MOTION BY SLIiJIKG CONTACT. 157 

number and decrease in size, because these conditions 

diminish the obliquity of action, as well as the amount of 

sliding. But in bevel wheels the action of the outer tooth 

outlines does not deviate much from the plane tangent to 

the two normal cones at P (Fig. 96) , and hence they act the 

same as spur wheels having the radii AP^ BP, which are 

larger than the radii of the bevel wheels themselves in the 

,. AP ,BP ^ , AP CP ^. BP CP . ^ 
ratios— and— . But — = -, and — = — . In 

other words, the action of a bevel wheel, so far as it is 
affected by the number of its teeth, is equal to that of a spur 
wheel of ilie same pitch whose radius is greater than that of 
the given bevel wheel in the same ratio that the slant height 
of the pitch cone is greater than its altitude. 

In a pair of mitre wheels this ratio is ly^, so that the action 
of a mitre wheel having, say, fifty teeth is equivalent to that 
of a spur wheel of seventy teeth. 

*144. Skew Bevel Wheels. — The theoretical construc- 
tion, as well as the practical manufacture, of the exact forms 
of teeth for skew bevel wheels, are both extremely compli- 
cated and laborious operations, and consequently they are 
rarely employed in practice. When skew bevel wheels are to 
be used, however, their teeth may be laid out by the following 
approximate method, which will give results abundantly ac- 
curate for all practical purposes. Having determined the 
pitch surfaces, as in Fig. 26, and decided on the frusta to be 
employed, we draw, at each end of each frustum, a cone 
normal to the respective hyperboloid. These cones are then 
to be developed, and teeth are to be laid out on them accord- 
ing to Tredgold's method. In this construction, it must be 
borne in mind that the relative numbers of teeth of the two 
wheels are not in the same proportion as the radii of the 
base circles, as in the case of cones ; for (Eq. 7, p. 40) these 
numbers are evidently proportional to the sines of the angles 
made by the projection of the common element with the pro- 



158 ELEMENTARY MECHANISM. 

jections of the respective axes, these projections being made 
on a plane parallel to the common element and both axes. 
The two wheels, then, have different circumferential pitches, 
and the ratio of the latter must be determined in each special 
case. The teeth having been laid out on the development of 
the normal cones, the latter are then to be replaced, the outer 
and inner cones being given the proper position with regard 
to each other by bringing the pitch points of a pair of corre- 
sponding tooth curves on the same generatrix. The surfaces 
of the teeth will be formed by joining the corresponding 
points of these curves by right lines. 

145. As an illustration of the method of Art. 47, let the 
axes be given as in Fig. 26, and let it be required to connect 

these axes so that —=3. Let DB' be taken as the interme- 

a 

diate axis, and let a' represent its angular velocity. Dividing 

r II r 

the fraction — into two factors, such as — = f and -^ = 2, 
a a a 

we have simply to solve two ordinary bevel wheel problems : 
first, to connect the axes BR' and DB' so that — = f ; sec- 

a 
r 

ond, to connect the axes DB' and B' 0' so that-^ = 2. As 

a 

DB' is perpendicular to both axes, the angles to be divided 
according to the method of Art. 38 are both right angles, so 
that the construction is very simple. 

If the axes pass so near each other that the common per- 
pendicular is too short to be used, some other line, such as 
HO^ must be taken. In this case, it becomes necessary to 
determine the true size of the angles PRO and POR ; and 
this is most conveniently done by revolving the line RO in 
turn about each of the axes until it is parallel to the vertical 
plane of projection, when the angle which it makes with the 
respective axis will be shown in its true size. 



MOTION BY SLIDING CONTACT. 159 

146. Axes neither Parallel nor Meeting- con- 
nected by Wheels with Involute Teeth. — In Fig. 74 
we have shown a pair of wheels with involute teeth, DE 
being the line of action. In the figure the wheels are in 
the same plane, and the point of contact is always situated 
in the line DE. 

The upper wheel remaining fixed, suppose the plane of the 
lower wheel to be revolved through any given angle about 
the line DE, as on a hinge. The two wheels will now lie in 
different planes, their axes being neither parallel nor inter- 
secting. The line DE will be the intersection of these two 
planes ; and the position of each wheel in its own plane, with 
reference to that line, is unaltered. But DE is the locus of 
contact ; and, as the position of neither wheel with reference 
to DE has been changed, it follows that the velocity ratio 
of the wheels will not be affected by the inclination of their 
planes. A¥hen the wheels are so inclined, they can, of 
course, move only in the direction which makes DE the locus 
of contact. If they are required to move in the reverse 
direction, they must be swung about a line similarly inclined 
to the line of centres in the opposite direction ; but it is evi- 
dent that in no case can they drive in both directions except 
when they are in the same plane. 

This property of involute teeth, of transmitting motion 
between axes neither parallel nor meeting, is only true when 
the wheels are very thin ; so that in practice the teeth of one 
wheel must be rounded so as to touch those of the other in 
points only, and not in lines. 

147. Face Gearing-. — Before the introduction of bevel 
gearing, the problem of transmitting motion between axes 
that were not parallel was usually solved by means of face 
gearing. Let two face wheels with cylindrical pins, exactly 
alike in every respect, be placed in gear, as shown in plan 
and elevation in Fig. 98, with their axes at right angles ; the 
latter not meeting in a point, but having their common per-. 



160 



ELEMENTARY MECHANISM. 



pendicular equal to the diameter of the pins. Then will 
these wheels revolve together with the same angular velocity. 




3^ig. 98 



Let B be the driver, and let the pins c, ^, be in contact. 
The distance between the axes of these pins is the sum of 
the radii of the pins ; that is, the diameter of a pin, or, what 
is equal to this diameter, the perpendicular distance between 
the axes of the wheels. 

Let the driver B turn, in the direction of the arrow, 
through one-sixth of a revolution ; the pin g moving to the 



MOTION BY SLIDING CONTACT. 161 

position e, and driving before it the pin c to the position b. 
The distance between the axes of the pins is equal to the 
diameter of a pin, as before ; and consequently the length 
of the perpendicular let fall from g on Be must equal the 
length of the perpendicular let fall from c on ^6. In other 
words, Bg sin gBe = Ac smhAc ; and, as Bg = Ac^ we have 
sin gBe = sin hAc. Hence angle gBe = angle bAc, which 
proves the equality of the angular velocities. 

The driver was in this case supposed to turn through an 
angle of sixty degrees ; but this was merely a matter of con- 
venience, as the same proof could have been applied to any 
other angle. The pin g must not be so long that its end will 
come into contact with the pin h, as the wheels revolve in the 
directions of the arrows. This consideration fixes the max- 
imum length of the pins, which is the same in both wheels. 

148. Axes Intersecting'. — As the common perpendic- 
ular to the two axes becomes less, the diameter of the pins 
decreases ; so that, when the axes intersect, the pins become 
mere lines. In order to transmit any power, the pins must 
manifestly have some thickness ; but the}" cannot be cylin- 
drical on both wheels. The pins on one wheel maj^, how- 
ever, still be cylinders, in which case the shape of those on 
the other may be found in the following manner. Suppose 
the axes of the wheels shown in Fig, 98 to be brought to- 
gether so as to intersect, the pins thus reducing to mere lines. 
Instead of having the corresponding pins in contact, as in 
Fig. 98, let the lower wheel be turned through a small angle, 
so as to separate the pins by some arbitrary distance, as 
shown in Fig. 99. Now, if both wheels be turned, in the 
directions of the arrows, with the same angular velocity, it is 
evident that the common perpendicular between any two cor- 
responding pins will change according to the positions of the 
pins at any instant ; and the length of this perpendicular 
can readily be determined for any positions of the pins. If, 
still using mere lines for the pins of the upper wheel, we 



162 



ELEMENTARY MECHANISM. 



now expand the pins of the lower wheel into solids of revo- 
lution, the radius of whose cross-section shall at any height 
be equal to this common perpendicular, it is evident that the 
two wheels will work together with a constant velocity ratio. 




ITig. 99 

If we now expand the mere lines, which act as pins of the 
upper wheel, into practical cylinders, their projections will 
become circles, and the curves for the lower pins will then 
be found by a process similar to that employed (Art. 122) in 
pin gearing. 

The principal advantage of face gearing is the facility of 
turning the pins and cogs in a lathe ; but on the other hand, 
we have the serious drawback, that the pressure between the 
pins is exerted at a single point only. 



MOTION BY SLIDING CONTACT. 



163 



149. Intermittent Gearing". — In Fig. 100 is shown 
the manner of connecting a pair of axes so that a uniform 
rotation of the one shall produce a given intermittent motion 
of the other. In such motions, the maintenance of an exact 




ITig. lOO 



velocity ratio is not usually essential, the important points 
being that the follower shall be caused to turn through a defi- 
nite angle, and then be securely held in its new position until 
it is again to be put in motion. 



164 ELEMENTARY MECHANISM. 

Suppose the axes ^4 and B to be given in position, and let 
it be required to connect them so that during every revolu- 
tion of the driver, the follower shall turn through ninety de- 
grees, the velocity ratio during the motion of the latter being, 

r 

as near as may be, — = f • The diagram shows the solution 
a 

of this problem, the pitch circles of the two wheels being 
drawn of the proper size to give the velocity ratio, as usual. 
The teeth are formed in the usual manner by means of the 
describing circles shown. The projections on the follower 
have the same outlines as the other teeth, except that they 
are longer, and their tops are connected b}^ an arc of the 
same radius as the smooth portion of the driver. 

Supposing the driver to move from its present position in 
the direction of the arrow, the smootli arc of the driver will 
slide along that of the follower, and no motion will be pro- 
duced in the latter until tlie point h of the driver reaches 
the line of centres AB. At this moment the point a of the 
driver's tooth will come into contact with the point h of 
the follower, and the latter will begin to move. The wheels 
will then continue in gear, the different teeth coming succes- 
sively into action, and motion being transmitted with the 

exact velocity ratio, — = f , until contact ceases between the 

a 
point d of the driver and the point e of the follower. These 
two points, at the moment of quitting contact, will be at the 
point n of the upper describing circle. As soon as these 
points have quit contact, the edge c of the driver will operate 
on the edge fg of the follower, turning the latter into the 
position now occupied by the edge hh ; and it will be held in 
that position until the point a again comes round into contact 
with g. The velocity ratio in this motion is exact while the 
driver moves through an angle aAd^ and the follower through 
an angle eBn, 



MOTION BY SLIDING CONTACT. 1^5 



CHAPTER IX. 

COMMUNICATION OF MOTION BY SLIDING CONTACT. 
VELOCITY RATIO AND DIRECTIONAL RELATION CONSTANT OR 

VARYING. 

Cams. — Endless Screw, — Slotted Link. — Whitworth's Quick Be- 
turn Motion. — Oldham^ s Coupling. — Escapements. 

150. In the last four chapters have been discussed the 
cases of sliding contact where both the velocity ratio and 
the directional relation were necessarily constant ; in the 
present chapter will be presented the various arrangements 
in which either or both of these may vary. 

151. A Cam is a plate which transmits motion to its fol- 
lower by means of its curved edge, or by means of a curved 
groove cut in the surface of the plate. When the motion 
is small or intermittent, such plates are often called tappets^ 
or loipers. 

In most cases which occur in practice, the conditions to be 
fulfilled in designing a cam or wiper do not directly involve 
the velocity ratio ; usually a certain series of definite posi- 
tions is assigned which the follower is to assume when the 
driver is in a corresponding series of definite positions. In 
cam motions, the motion of the follower is usually derived 
from the cam by means of a cylindrical roller turning about 
a smaller pin as an axis, the latter being rigidly fastened to 
the follower. This has the advantage that nearly all the 
wear is concentrated on this axis, which may readily be 



166 ELEMENTARY MECHANISM. 

renewed when worn out. If the pin is to be driven by the 
cam in one direction only, being made to return by the force 
of gravity or the elastic force of a spring, the cam need only 
have one acting edge ; but if the cam is to drive the pin in 
both directions, it must have two acting edges, with the pin 
between them, so as to form a groove or a slot of a uniform 
width equal to the diameter of the pin, with clearance just 
sufficient to prevent jamming or undue friction. 

The centre of the pin may be treated as practically at all 
times coinciding with the centre line of such a groove, which 
centre line may be called the pitch line of the cam. 

The most convenient way to design a cam is usually to 
draw in the first place its pitch line, and then to find the 
acting edge or edges by the process of Art. 122 ; using a 
radius slightly greater than that of the pin in case two 
edges are employed. 

152. Construction of the Cam Curve. — In Fig. 101, 
let A be the centre of motion of the proposed cam, and BH 
the path of the centre of the pin on the follower ; the con- 
dition being that the centre of the pin shall start from the 
point H, and assume in succession the positions E, D, (7, 
and B while the cam revolves throus-h successive anoles of 
thirty degrees. With centre A and radius AH, describe the 
circle HN; produce the radius ^^ to S, and draw the other 
radii (produced) , AK, AL, AM, and AN, at successive angu- 
lar intervals of thirty degrees. With centre A, draw circular 
arcs through the successive positions E, D, C, B, of the pin, 
and on these arcs lay off the distances Kk = Cc, LI = Dd, 
Mm = Ee. Then will k, I, and m be points of the cam 
curve required. The curve nmlkB, drawn through these 
points and N and B, will be the curve which will fulfil the 
required conditions ; for, assuming n to be at H, and the 
cam to revolve in the direction of the arrow, it is evident 
that as the radii AM, AL, AK, and Ah successively come 
into the position AS, the joints m, I, k, and B of the cam 



MOTION BY SLIDING CONTACT. 



167 



curve will coincide witli E^ Z), C, and B respectively, thus 
driving the pin as required. To find the curve for a pin of 
sensible diameter, we proceed as in Art. 122, drawing circles 
of the same diameter as the pin in a sufficient number of 
positions along the pitch line already found, and then draw- 
ing the acting edge tangent to these circles. 




When the path of the pin passes through the centre of 
motion of the cam, the distances Ee^ Del, etc., all reduce to 
zero ; and the pitch line is drawn through the points of inter- 
section of the successive radii and the circular arcs through 
the corresponding positions of the pin. • 

As the angle BHS increases, the action between the edge 
of the cam and the pin becomes more oblique, thus increasing 
the friction ; and it is hence advisable to make that angle as 
small as possible ; in other words, the path of the pin should 
point as near as possible to the centre of motion of the cam. 

In case the motion of the follower is required to be uni- 



168 



JlLEMENTARY MECHANISM. 



form, the distances HE, ED, DC, and CB would all be 
equal, but no modification of the method of construction 
would thereby be introduced. 

153. Another Example. — In Fig. 101 the path of 
the follower is a straight line, and the cam has uniform 
motion about a fixed centre. But none of these conditions 




need be adhered to. The path of the follower may be any 
curve whatever, and it may move in this path in either direc- 
tion, and with uniform or varying velocity. The cam usually 
revolves about a centre, or has rectilinear motion ; but its 
velocity may also be varied at pleasure. All these possible 
variations give rise to an endless variety of shapes for the 
cam curves, but the principles underlying their construction 
are always the same. Thus, in Fig. 102, let the path of 



MOTION BY SLIDING CONTACT. 169 

the pin be the curved line HB^ and let the pin successively 
occupy the positions jP, E^ Z), etc., while the cam revolves 
in the direction of the arrow through the unequal angles 
NA3f, MAL, LAK, etc. The radii being drawn at the 
given angles, circular arcs are drawu through F, E, D, etc., 
and the points of the curve found, just as in Art. 152, by 
making Rh = Cc, Kk = Z)fZ, etc. \ 

*154. Cam for Complete Revolutions. — In Figs. 101 
and 102 the directional relation is constant ; in other words, 
the direction of rotation of the cam must be reversed in 
order to bring the pin down again to H. But this may be 
accomplished by simply adding to the curve of the cam, in 
which case the latter may revolve continually in the same 
direction. The law of motion of the pin in one direction 
may be entirely different from that in the other direction, 
and the pin may be given an interval of rest at any eleva- 
tion by making the corresponding part of the cam curve an 
arc of a circle. 

In Fig. 103, let A be the centre of motion of the cam, and 
let the vertical numbered line be the path of the follower. 
The cam is to revolve uniformly at the rate of one revolution 
in twelve seconds. 

Each number on the vertical line shows the required posi- 
tion of the pin at the end of the second indicated by that 
number. 

Draw twelve equidistant radii, and draw circular arcs 
through the various positions of the follower. Making 
la = la', lib = 2b% Illf = 3/, IVh = Ak', etc., we 
find the points of the curve, as before. The interval of 
rest indicated by the coincidence of the numbers 7 and 8 is 
obtained by means of the circular arc cc. The cam in the 
figure is drawn in the position when the pin is at the point 
012; and the cam is ready, by one complete revolution 
in the direction of the arrow, to cause the pin to go tb^^^ugh 
the cycle of motion required. 



170 



ELEMENTARY MECHANISM. 



* 



mig. 1.03 




MOTION BY SLIDING CONTACT. 



171 



155. Cam moving in Straight Path. — In all the pre- 
ceding cases we have assumed the cam to revolve about some 
fixed centre of motion. But this is not a necessary con- 
dition ; it may move in any path whatever. In practice, 
however, there is but one other path employed; viz., the 
straight line. In Fis;. 104, let ABCD be a flat rectangular 




Fig. 104r 



plate moving in the direction of its length, and let the path 
of the follower be the line MN, at right angles with the 
direction of motion of the plate. Let the pin of the follower 
start from a position of rest at P, and move with a gradually 
accelerated velocity, so that it occupies the positions 1, 2, 3, 
4, etc., at the end of equal successive intervals of time. 
Lay off on AB the distances MI, I II, II III, III IV, etc., 
through which the plate moves during the same intervals. 
In the figure the plate is supposed to be moving uniformly, 
and these distances are consequently equal ; but they may 
vary according to any assigned law. Draw the lines la, 
lib, IIIc, IVd, etc., parallel to 3IN, and the lines la, 2&, 
3c, id, etc., parallel to AB. Their intersections, a, h, c, d, 
etc., will be points of the required cam curve. From this 
the working curves are derived in the usual manner, being the 



1T2 ELEMENTARY MECHANISM. 

sides of a groove or slit of the proper width in the plate. 
The theoretical curve above found is the centre line of the 
groove. 

In the special case shown in the figure, the lines Ja, ia, 
Ilh^ 2b, etc., are at right angles; the angle between them 
is always the angle between the directions of motion of the 
plate and the follower. If in Fig. 104 we make the velocity 
of the follower also constant, the cam curve will become a 
straight line ; for instance, if we assume that the follower is 
to traverse the distance P7, with a uniform velocity, during 
the same time that the plate moves, also with a uniform velo- 
city, over the distance M VII = Ig, then the straight line 
PG must be the cam line required. This line will evidently 
be the hypothenuse of a right triangle, the other two sides 
of which are the lines representing the respective distances 
traversed by the plate and the follower in the same time. 
The velocity ratio of the cam plate and the follower in this 
special case is evidently constant, and is simply the ratio of 
the isochronous distances above mentioned. 

156. The Screw. — If a plate with a straight slit or 
groove, as just described, be wrapped around a cylinder 
whose axis is parallel to the path of the follower, the slit in 
the plate will become a spiral groove in the cylinder. If the 
cylinder be revolved uniformly, this groove will impart pre- 
cisely the same motion to the follower as before. 

If the length of the plate be greater than the circumference 
of the cylinder, the spiral groove will encompass its surface 
through more than one convolution, and may in this way 
proceed in many convolutions from one extremity of the 
cylinder to the other. Such a recurring spiral is called a 
scretv. The inclination of the spiral to a line drawn on the 
surface of the cylinder parallel to the axis is constant, and 
is the same as tlie inclination of the straight line in the flat 
plate to the path of the follower. 

The pitcJi of a screw is the distance between successive 



MOTION BY SLIJ3ING CONTACT. ITS 

convolutious of the spiral measured along a rectilinear ele- 
ment of the cylindrical surface. 

The screw is sometimes made in this elementary form, 
consisting of a simple spiral groove which gives motion to 
a slide, by means of a pin fixed to the latter, and lying in 
the groove ; but generally screws receive a more complex 
arrangement. 

In the first place the pitch is made comparatively small, 
the necessary motion of the follower being secured by a 
corresponding increase in the number of revolutions of the 
screw. The convolutions of the groove are brought so 
close together that the ridge which separates two contiguous 
grooves becomes the counterpart of the groove itself. This 
ridge is termed the thread of the screw ; and from its section 
the screw derives its distinctive title, such as square- threaded, 
V- threaded, and round- threaded. 

In the second place, instead of a single pin, other pins 
may be fixed to the follower opposite the other convolu- 
tions ; then, since each pin will receive an equal velocity 
from the revolving cylinder, the motion of the follower will 
be effected as before, with the advantage of an increased 
number of points of contact. But this series of pins may 
be replaced by a short comb or rack, the outline of which 
exactly fits that of the threads of the screw. This is the 
most ancient form in which the screw was employed. 

Most commonly, however, the piece which receives the 
action of the screw is provided with a cavity embracing the 
screw, and fitting its thread completely ; being, in fact, a 
hollow screw corresponding in every respect to the solid 
screw. Such a piece is termed a nut^ and the hollow screw 
an inside screio^ the solid screw being then called an outside 
screiv. These modifications are only introduced to distribute 
the pressure of the screw upon a greater surface ; for, as the 
action of the thread is exactly alike upon every section of 
the nut, the result of all these conspiring actions is the same ; 



174 ELEMENTARY MECHANISM. 

namely, that the piece to which the pin or comb or nut is 
attached advances in a direction parallel to the axis of the 
screw through a distance equal to the pitch for every revolu- 
tion. 

157. A screw may be right-handed or left-handed; the 
majority of screws are the former, the latter being used 
only when other conditions make it necessary. Supposing 
the nut to be fixed, a right-handed screw will enter its nut 
when turned in the direction of the hands of a clock ; a left- 
handed screw must be turned in the opposite direction. 

If the inclination of the thread of a screw to the recti- 
linear elements of the cylinder be very great, one or more 
intermediate threads may be added. In such cases the screw 
is said to be double-threaded^ triple-threaded^ etc., according 
to the number of separate spiral threads on the cylinder. 

Screws whose pitch is an aliquot part of an inch are 
usually classified by mentioning the reciprocal of the pitch ; 
i.e., if a certain screw has a pitch of one-quarter of an inch, 
it is spoken of as having four threads to the inch. 

During one complete revolution of any screw, the follower 
will evidently move through a space equal to the pitch of the 
screw; i.e., through a space equal to the distance between 
successive convolutions of the same spiral measured on a 
rectilinear element of the screw cylinder. 

When the comb or rack form above spoken of is used, 
the screw is frequently made short, and the rack lengthened. 
If it is essential that the screw shall always remain com- 
pletely in gear with the rack, then the maximum length of 
path described by the latter will be the difference between 
their lengths. 

158. Endless Screw. Worm and Wheel. — From 
the rack driven by a short screw, we readily pass to the so- 
called endless screw ^ shown in Fig. 105. In this combina- 
tion the screw, or ivorm^ BB, gives motion, not to a rack, but 
to the wheel C. The screw is so mounted that it can have 



MOTION BY SLIDING CONTACT. 



175 



no motion except that of rotation, and the wheel has teeth 
of the same pitch as the screw thread. If the screw axis 
be turned around, every revohition will cause one tooth of 
the wheel to pass across the line of centres ; and as this 
action puts no limit, from the nature of the contrivance, to 
the number of revolutions in the same direction, a screw 
fitted up in this manner is termed an endless screw, in oppo- 
sition to the ordinary screw, which, when turned around a 
certain number of times either way, terminates its own action 
by bringing the nut to the end of the thread. 




Fig. 105 



159. Shape of the Teeth If we make any meridian 

section of the screw, we will find it to be a rack ; in fact, the 
screw may be moved in the direction of its length, and will 
then drive the wheel precisely in the manner of a rack. 
Consequently, if the wheel be merely a thin plate, we need 
only make the meridian section of the screw a rack with 
teeth laid out in the usual manner to work correctly with the 
assumed wheel tooth. But as in practice the wheel must 



176' ELEMENTARY MECHANISM. 

be given some thickness, it is necessary to determine the 
proper form of the teeth for that case. 

If we make a series of sections through D, E^ etc., parallel 
to the mid-plane of the wheel, we will still find the section 
of the screw to be a rack, though the outlines of the teeth 
will change in shape in each section. 

If, now, we make the outline of any section of the wheel 
tooth of the proper shape to gear correctly with the outline 
of the corresponding section of the screw tooth, considered 
as the tooth of a rack, we shall evidently have a point of 
contact between the corresponding teeth in every section. 

In fact, if we make a wheel tooth whose shape is continu- 
ally changing in every section to correspond to the change in 
the same section of the screw tooth, we shall have the teeth 
in contact at each instant along a line, so that the wear will 
be distributed along a surface. Such a screw is called a 
close-fitting or tangent screw. 

160. Practical Method of Ciittinur Wheel Teeth. — 
The practical difficulty of making the teeth of a wheel of 
which the form in every parallel section shall be different, is 
very simply overcome by making the screw cut the teeth. 

An exact copy of the tangent screw is made of steel, the 
edges of its threads are notched, and it is then hardened, so 
that it becomes a cutting tool. It is then mounted in a suit- 
able frame, so as to gear with the roughly formed teeth on 
the wheel, and turned so as to drive them ; in the course of 
which operation it cuts them to the proper figure. The axis 
of the cutting screw is placed at first at a distance from the • 
axis of the wheel somewhat greater than the intended per- 
manent distance ; and, after each complete revolution of the 
wheel, the axes are brought nearer together, until the per- 
manent distance is attained ; and, by turning the screw in 
this last position, the shaping of the teeth is finished. An 
involute wheel tooth working with a screw tooth whose 

sloping sides, is the best 



MOTION BY SLIDING CONTACT. 177 

combination, as the successive diminutions of the distance 
between the axes will not affect the velocity ratio (Art. 108). 
In order to secure a good arc, of action, and diminish obliq- 
uity, such wheels should not be given less than about thirty 
teeth. 

In order to avoid weak corners in the wheel teeth, their 
sides are usually bounded by straight lines, BH and BK, 
radiating from the axis of the worm ; and the angle HBK 
usually varies between sixty aud ninety degrees. 

161. Hour-Glass Worm Instead of making the pitch 

surface of the worm a cylinder, we may make it conform to 
the curvature of the wheel. In that case its pitch surface 
will be the surface produced by revolving an arc of the 
wheel pitch circle about the axis of the worm, thus forming 
the shape from which the worm derives its name. This 
arrangement is also named, after its inventor. Hind ley's 
screw. The acting surfaces of both the worm and the 
wheel are very peculiar ; but the arrangement may, neverthe- 
less, be very easily constructed in practice. 

Just as in the ordinary tangent screw, we must first pre- 
pare a cutting screw. To obtain this, a tool whose cutting 
edges are formed in the shape of the proposed wheel tooth 
is so clamped to a horizontal revolving plate of the size of 
the proposed wheel that the plane of its cutting edges passes 
through the axis of the worm. The plate and the worm 
blank being rotated at their proper relative velocities by 
means of some interposed mechanism, the distance between 
the two axes is gradually diminished until the permanent dis- 
tance is reached, during which operation the worm will be 
cut to the proper shape. By taking such a worm, notching 
its edge to make a cutting tool of it, the wheel teeth can 
then be cut just as in the ordinary worm and wheel. 

Such teeth are in contact along a line in the meridian plane 
of the screw, but do not come in contact along the whole 
surface of a wheel tooth. 



178 ELEMENTARY MECHANISM. 

162. The endless screw falls under the case of two 
revolving pieces whose axes are not parallel and do not 
meet. It communicates motion very smoothly, and is equiv- 
alent to a wheel of a single tooth, because one revolution 
passes one tooth of the wheel across the line of centres ; but, 
generally speaking, it can be employed only as a driver, 
on account of the great obliquity of its action. A worrn 
may be multiple-threaded, just like any other form of screw, 
and, in that case, will pass as many wheel teeth across the 
line of centres for every revolution of the worm as there are 
separate threads on the latter. The practical process of 
cutting the teeth is, however, the same as before. 

163. Screw to produce Variable Motion. — In all 
the cases previously described, the screw has been supposed 
to have a uniform pitch, and hence to produce a uniform 
motion in the follower. But we may impart any motion 
whatever ; the only condition being, that the pitch of the fol- 
lower must not deviate much from a straight line parallel to 
the axis of the screw. As the inclination of the spiral 
groove varies, the velocity of the follower changes ; a period 
of rest of the follower being obtained by making the inclina- 
tion zero. A small intermittent motion may readily be 
obtained by making the groove in the shape of a simple ring, 
except at a certain portion, where it deviates the necessary 
amount. 

If it be required that the follower shall move back and 
forth while the screw revolves continually in the same direc- 
tion, the spiral must be cut in both directions ; in which case 
the follower cannot be a rack or nut or wheel, but must be 
a single pin or similar piece. On the cylinder of the screw 
are cut two complete spirals, one right-handed and the other 
left-handed, joined together at their ends ; so that the two 
screws form one continuous path, winding around the cylinder 
from one end to the other and back again continuously. 
When the cylinder revolves, the piece which lies in this 



MOTION BY SLIDING CONTACT. 179 

groove, and is attached to the follower, will be carried back- 
wards and forwards ; and each total oscillation will corre- 
spond to as many revolutions of the cylinder as there are 
convolutions in the compound screw. As the screw grooves 
necessarily cross, each other, the piece that slides in them 
must be made long, so as to occupy a considerable length of 
the groove ; thus making it impossible for it to quit one screw 
for the other at the crossing places. Also, as the inclina- 
tions of the two screws are in opposite directions, it is neces- 
sary to attach that piece to the follower by means of a pivot, 
so as to allow it to turn through a small arc as the inclina- 
tion changes. By varying the inclination at different points, 
the velocity ratio may be varied at those points. 

*164. Piu and Slotted Crank. — In Fig. 106, let A be 
the centre of rotation of an arm, AP, carrying at its extrem- 



ity a pin, P, which slides freely in the slot in the piece BG. 
The latter has its centre of rotation at B. If the arm AP 
be revolved uniforml}^, it will impart a variable velocity to 
the arm BC. 

Let Pa, perpendicular to AP, represent the linear velocity 
of the pin in the circle MJSF. Draw an indefinite line per- 



180 ELEMENTARY MECHANISM. 

pendiciilar to PB at P, aDcl let fall on it the perpendicular 
ah ; then will Ph be the linear velocity of the point P of the 
arm BC at that instant. Let a = angular velocity of the 
arm AP^ and a = angular velocity of the arm BG. Also 
let the constant length AP be designated by i?, and the 
variable length BP by r. 
Then 



But 

hence 



Pa , Ph 



Ph = Pa cos uPh = Pa cos APB, 



Pa cos APB -.a' P . „„ 
, and — = — cos APB, 



"When APB = 0, that is, when both the arms lie in the 
line of centres DE, the limiting values of the velocity ratio 

will be obtained. When P is at E, the velocity ratio — has 

a 

P 7? 

its maximum value, — = --. The ratio becomes 

r B - AB 

smaller as P leaves E and approaches D. at which point 

— has its minimum value, — = -— -— . 

a ' r E + AB 

So long as AB is less than P, we may by this means 
cause one arm to revolve with a variable velocity by means 
of another arm revolving uniformly. 

When AB exceeds P, the second arm merely swings on 
each side of the line of centres through an angle whose sine 

is — — . When it is at the end of its outward swing, the 
AB 

angle APB = 90° and - = - x = ; showing that for 
a r 

that instant no motion is imparted to the arm BC by the 

rotation of AP. 



MOTION BY SLIDING CONTACT. 181 

The necessary length of slot when B lies within the circle 
MN is the diameter of the pin + BD — BE ; when B lies 
without MN^ the necessary length of slot is the diameter of 
the pin + 2AP. 

165. Wbitwortli's Quick Return Motion If in 

Fig. 106 we attach a connecting-rod to the end C of the 
arm BC^ and compel the other end of the rod to move in a 
straight line perpendicular to AB at B^ we will have a com- 
bination such as is represented by Fig. 107. The length of 
the stroke is evidently 2BC. If the arm AP be revolved 
uniformly, the forward and back strokes of Q will be made in 
times proportional to the arcs adh and acb. We thus have a 
form of "quick return" motion. This has been applied in 
a modified form, as shown in Fig. 108, to a shaping machine, 
by Sir J. Whitworth. 




Fig. lor 



In the figure, i) is a plate spur wheel which turns about its 
centre A^ upon the large fixed shaft. A pin, P, fixed in and 
projecting from the face of wheel Z>, corresponds to the 
point P in Fig. 107, so that AP is the arm which revolves 
uniformly. A pin, -B, eccentric in the large shaft, is the 
centre about which the arm of varying length turns ; BP qoi^ 



182 ELEMENTARY MECHANISM. 

responding to BP of Figs. 106 and 107. A crank piece, E^ 
turns about 5, and has a slot in one end, in which P shdes. 
To the opposite end of this piece the connecting-rod is 
attached. The end Q of the rod is attached to the sliding 
head which carries the cutting tool. 



Fig. 108 




As D revolves, motion is given to Q by means of the pin 
P and the crank piece, and the varying distance. of P from 
B exactly replaces the arm of varying length. The length 
of stroke is adjusted by altering the position of C in that 
end of the crank piece, thus changing the length of the crank 
arm J5(7, but in no way affecting the ratio of the periods of 
advance and return. Thus, for example, if the arc acb (Fig. 
107) is one-third of the circumference, adh being two-thirds, 
the period of advance is to the period of return as 2 is to 1, 
without regard to the actual length of stroke. 

166. Pin and Slotted Sliding^ Bar. — In Fig. 109, let 
the pin P be fixed at the extremity of the uniformly revolv- 
ing arm AP^ as before. The piece B is free to move in the 
direction CD or DO only, and has in it a slot perpendicular 
to the line DC^ in which the pin slides. Let Pa = V == 
linear velocity of the pin in the arc of the circle ; then v, 
equal to linear velocity of the piece B in the direction AC, 
will be found, as in the last article, by dropping the perpen- 
dicular ah on the line -P6, the latter being perpendicular to 
the line of the slot. 



MOTION BY SLIDING CONTACT. 



183 



Hence 



hence 



V Pb . r» 7 • -n A-n 

V Pa 



V = FsinP^^. 



When PAB is or 180°, i.e., when P is at i) or E, we 
have V = 0; when PAB = 90° or 270% v = V. 

This motion of P, varying between and V, and going 
from to F and from F to twice in each revolution of AP, 
is called harmonic motion. 




The necessary length of slot is 2AP -f- diameter of pin. 
The length of the path of B is 2AP. 

This arrangement is much used in some varieties of pumps 
to connect the fly-wheel shaft with the piston rod. 

167. Cam and Slotted Sliding Bar. — In Fig. 110 
is shown an example of a cam which, by its uniform rota- 
tion, produces a motion similar to that, of Fig. 109, but with 
intervals of complete rest. The cam consists of a triangular 
piece ; the sides of the triangle being three equal arcs, each 
described around the point of intersection of the other two. 
The cam revolves uniformly about one of its corners, as A: 



184 



ELEMENTARY MECHANISM. 



The follower is the slotted bar BB^ and the cam acts upon 
the two straight edges of the slot, the distance between 
which is equal to the radius of curved edges of the cam. 

Consequently the slot will be in contact with an angle and 
a side of the cam in every position, and the motion produced 
is as follows : Let the circle described by the outer edge of 
the cam be divided into six equal parts, as in the figure. 
Tracing the motion as the angle m of the cam goes round 
the circle in the direction of the numbers, it appears that no 
motion will be given to the bar while m is moving from 1 to 
2. While m travels from 2 to 3, the face Am drives the 
upper side of the slot with an increasing radius ; and hence 
the bar begins to move, and its velocity gradually increases. 
While m travels from 3 to 4 the action is similar to that of 
Fig. 109, and the motion of the bar will gradually be 
decreased until m reaches 4, when the bar will come to rest. 




As m moves from 4 to 5 the bar remains at rest ; from 
from 5 to 6 the bar begins to move with an increasing 
velocity ; from 6 to 1 the bar moves with a decreasing 
velocity, coming to rest as m reaches 1. 

168. In case the direction of motion of the follower 
intersects the axis of motion of the cam, the latter may be 
made in the shape of a screw thread on a cone ; wlien the 



MOTION BY SLIDING CONTACT. 



185 



follower's direction neither intersects nor is parallel to the 
cam axis we may employ a screw thread on a hyperboloid 
of revolution. 

In fact, almost any kind of motion may be ol)tained by 
means of a suitably shaped cam ; but the general principles 
employed in the various cases above treated of apply equally 
well to any other special cases. 

169. Oldham's Coupling' In Fig. Ill is shown a 

method of communicating equal rotation by sliding contact 
between two axes whose directions are parallel. Aa and Bb 
are the axes, each of which is furnished with a forked end, 
terminated by sockets bored in a direction to intersect the 
respective axes at right angles. The whole is so adjusted 
that all four sockets lie in one plane perpendicular to both 
axes. A cross with straight cylindrical arms is fitted into the 
sockets in the manner shown in the figure, and its arms are 
of a diameter that allows them to slide freely in their respect- 
ive sockets. If one of the axes be made to revolve, it will 
drive the other with the same angular velocity. 




Fig. Ill 



For let the sketch at the right be a section through the 
cross perpendicular to the two axes Aa and Bb^ and let the 
large circles be those described by their respective sockets. 
Then, if O be a socket of Aa, the arm of the cross which 
passes through it must meet the centre A ; and in like man- 



186 



ELEMENTARY MECHANISM. 



ner, if 7) be a socket of Bb^ the arm DB must pass through 
the centre B. Also, if C move to C\ the new position 
(clotted lines) of the cross will be found by drawing C'A 
through A^ and BD' perpendicular to it through B. And it 
is evident that the angle C'AC = angle D'BD^ hence the 
angular velocity is the same in both axes. In every position . 
of the cross we will have the triangle APB^ in which the side 
AB is constant, and the angle APB opposite to it is always 
a right angle. Hence the locus of P must be the circle whose 
diameter is AB] i.e., the centre of the cross will travel 
around the small dotted circle whose diameter is the distance 
between the axes. Also every arm will slide through its 
socket and back again during each revolution through a space 
equal to twice the distance between the axes. 

In practice this arrangement is usually made in the shape 
of two discs, with a bar sliding in a diametral slit in each ; 
the two bars being rigidly connected in the form of a cross. 

170. An Escapenient is a combination in which a 
toothed wheel acts upon two distinct pieces or pallets attached 




I^ig. lis 



to a reciprocating frame, so that, when one tooth ceases to act 
on the first pallet, a different tooth shall begin to act on the 
second pallet. A simple example is shown in Fig. 112. 
The wheel A revolves coutinually in the direction of the 



MOTION BY SLIDING CONTACT. 187 

arrow. The frame lias two pallets, d aud e, and can only 
move ill the direction of its length. In the position shown, 
the tooth a is just escaping from the tooth cZ, and h is just 
ready to come in contact with e, by w^iich the frame will be 
driven to the left. The shapes of the teeth may be designed 
as usual for a wheel and rack, and the point of quitting con- 
tact is found by the intersection of the addendum line of 
the wheel teeth with the describing circle of the pallets. 
The number of teeth on the wheel must evidently be odd. 

But the frame may be used as the driver, instead of the 
wheel, by moving it alternately in each direction. This will 
cause the wheel to revolve in the opposite direction to that in 
which it would itself produce the reciprocation of the frame. 
But, when the frame is the driver, there is alwa^^s a short 
interval at the beginning of each stroke, during which no 
motion will be given to the wheel. 

171. Crown-wheel Escapement. — The crown-wheel 
escapement is used for causing the vibration of one axis by 
means of the uniform rotation of another. The latter carries 
a wheel consisting of a circular band, with large teeth, like 
those of a saw, on one edge. The vibrating axis, or verge^ 
as it is often called, is located immediately above the crown 
wheel, and in a plane at right angles to the wheel axis, the 
latter being vertical. The verge carries two pallets, project- 
ing from it in directions at right angles, and a sufficient 
distance apart so that they may engage alternately with teeth 
on opposite sides of the wheel. By this alternate action a 
reciprocating motion is set up in the verge. The rapidity of 
this vibration depends largely on the inertia of the verge, 
which may be adjusted by attaching a suitably weighted arm 
to the latter. 

This escapement, though but rarely used at the present 
day, is of interest as being the first contrivance used in a 
clock for measuring time. 

172. Anchor Escapement. — In Figs. 113 and 114 are 



188 ELEMENTARY MECHANISM. 

shown two forms of this ese:i[)em(Mit. In Fig. 118 the wheel 
has long, slender teeth, and turns in the direction of the 
arrow. The vibrating axis B carries a two-armed piece 
having pallets (J and D at its extremities, and resembling 
somewhat the form of an anchor, whence the name of the 
combination. When the tooth g presses against the pallet C\ 
the normal at the point of contact passes on the same side 
of the centres A and B ; hence (Art. 30) the tooth will tend 




Fig. 113 

to turn the pallet in the same direction as the wheel. BC 
will therefore turn upwards, and allow the tooth to escape 
from the pallet. At this instant the tooth k will begin to 
act on the pallet D ; and, as the normal here passes between 
the centres A and jB, BD will move in opposite direction 
to the wheel, and hence the tooth h will escape. 

The teeth in an anchor escapement are often replaced b}^ 
pins, in wdiich case the form of the anchor may be so altered 
that the action shall take place entirely on one side of the 
line of centres, as shown in Fig. 114. The rapidity of vibra- 
tion is controlled by the inertia of a weight or pendulum 



MOTION BY SLIDING CONTACT. 



189 



attached to the verge. This very inertia, however, prevent- 
ing the verge from being suddenly stopped and reversed in 
direction, causes a recoil action to be set up in the wheel, 
which materially diminishes the utility of this escapement ; 
for it is evident that, as the verge cannot be stopped sud- 
denly, the wheel must of necessity give way and recoil at the 
first instant of each engagement between a tooth and its cor- 
responding pallet. The greater the inertia due to the load 
attached to the verge, the more slowly will the escapement 
work, and the greater will be the amount of recoil. 




Fig. iifi 



173. Method of connecting' Anchor and Pendu- 
lum. — There is one uniform method of connecting the 
anchor and the pendulum, which can be seen in almost any 
clock. The pendulum, consisting often of a compound 
metal rod with a heavy bob, is swung by a piece of flat steel 
spring, and vibrates in a vertical plane very near to that in 
which the anchor oscillates. To the centre of the anchor is 
attached a light vertical rod, having the end bent into a hori- 
zontal position, and terminating in a fork which embraces 
the pendulum rod. It follows that the anchor and the pen- 
dulum swing together, though each has a separate point of 
suspension. 



190 ELEMENTARY MECHANISM. 

1 74. Action of Escapement on Pendulum. — In 

Fig. 113, let tlie escape wheel tend to move in the direction 
of the arrow, so as to press its teeth slightly against the pal- 
lets of the anchor ; the pendulum being hung from its point 
of suspension by a thin strip of steel, and vibrating with the 
anchor in the manner already stated. Let the arc abecd be 
taken to represent the arc of swing of the centre of the bob 
of the pendulum. As the pendulum moves from d to &, the 
point g of the escape wheel rests upon the oblique lower sur- 
face of the pallet (7, and presses the pendulum onward until 
the latter reaches 6, when the point of the tooth escapes at 
the end of the pallet. For an instant the escape wheel is 
free ; but a tooth is caught at once upon the opposite side 
by the oblique upper surface of D, and the escape wheel then 
presses against the pendulum, and tends to stop it, until 
finally the pendulum comes to rest at the point «, and com- 
mences the return swing. During the latter the pendulum 
is similarly at first urged on, and then held back by the 
action of the escape wheel. 

This alternate action unth and against the pendulum pre- 
vents the pendulum from being, as it should be, the exclusive 
regulator of the speed of revolution of the escape wheel ; for 
its own speed, instead of depending solely on its length, will 
also depend on the force urging the escape wheel round. 
Hence any variation in the maintaining force will disturb the 
rate of the clock. 

1 75. Dead-beat Escapement. — This objectionable 
feature is obviated in Graham's dead-beat escapement. Fig. 
115. It is, however, most worthy of note that the change in 
construction which abolishes the defects due to the recoil, 
and gives the astronomer an almost perfect clock, separates 
the combination entirely from its original conception; viz., 
that of an apparatus for converting circular into reciprocating 
motion. The improvement consists in making the lower 
surface of the pallet C and the upper surface of the pallet D 



MOTION BY SLIDING CONTACT. 



191 



arcs of circles, whose centre is at B, The oblique surfaces 
^m, np^ complete the pallets. As long as the tooth is resting 
on the upper surface of i), the pendulum is free to move, 
and the escape wheel is locked ; hence in the portion ha of 
the swing, and back again through a6, there is no action 
against the pendulum except the very minute friction which 
takes place between the tooth of the escape wheel and the 
surface of the pallet. Through the space he the point of 
the escape wheel tooth is pressing against the oblique edge 
nX>, and is urging the pendulum forward. 




:Pig. 115 



Then at c this tooth escapes, and the tooth upon the oppo- 
site side falls upon the lower surface of (7, and the escape 
wheel is locked ; from c to c?, and back again from d to c, 
there is the same friction which acted through ha and ah. 
From c to 6 the point of a tooth presses upon gm^ and urges 
the pendulum onward ; at h this tooth escapes, another one 



192 ELEMENTARY MECHANISM. 

comes into contact, and so on. It follows that there is no 
recoil, and the only action against the pendulum is the very 
minute friction between the teeth and the pallets. The term 
"dead-beat" has been applied because the seconds hand, 
which is fitted to the escape wheel, stops so completely when 
the tooth falls on the circular portion of the pallet. There 
is none of that recoil or subsequent trembling which occurs 
in the other escapements. 



MOTION BY LINKWORK. 193 



CHAPTER X. 

COMMUNICATION OF MOTION BY LINKWORK. 

VELOCITY RATIO AND DIRECTIONAL RELATION CONSTANT OR 

VARYING. 

Classification. — Discussion of Various Classes. — Qidck Return 
Motion. — Hookers Coupling. — Intermittent Linkwork. — Ratchet 
Wheels. 

176. As has been shown by the general definition (Art. 
22), linkwork derives its name from the rigid connecting 
piece or link. This connecting piece is known by various 
names under different circumstances, such as connecting-rod^ 
coupling rod^ side rod, eccentric rod, etc. The arms are 
known as cranks when they perform complete revolutions ; 
and as beams, crank arms, rocker arms, or levers, when they 
oscillate. 

177. Classification of liinkwork. — Linkwork is used, 

I. To transform circular motion into rectilinear reciproca- 
tion, or the reverse. 

II. To transform continuous rotation into rotative recipro- 
cation, or the reverse. 

III. To transmit continuous rotation. 

Examples of the first class are seen in slotting and shap- 
ing machines, power pumps, and in the usual forms of the 
steam engine ; of the second class, in steam engine valve 
motions, where a rocker shaft is employed ; and of the third 
class, in locomotive side rods. 



194 



ELEMENTARY MECHANISM. 



Class I. 

Transformation of Circular Motion irdo Rectilinear Reciprocation, 
and the Reverse. 

178. In Fig. 116, let AP be a crank revolving about the 
fixed centre A, and connected by a link PQ to a point Q, 
travelling in a straight line KL whose direction passes 
through the centre A. Let AP = E, and PQ = I. The 
length of the path of Q is evidently equal to 2R. When P 




is at C or Z), the points A^ P, Q, will be in one straight line. 
The points C and D are called dead points ; since when P is 
at either of them, the revolution of AP will cause no motion 
whatever to be transmitted to the point Q, for that instant. 
"When PQ overlaps AP^ as when P is at D, we shall term 
the point D the inward dead point ; and when Q lies at the 
other extremity of its stroke, so that P is at O, we shall 
term the point C the outward dead point. In Fig. IIG, let 
fall from P the line PE perpendicular to AQ. Then the 
distance of Q from A is at any instant, AQ = QE -\- AE 
= \P — P^ sin^ ■}- M cos ^, the last term of which will be 
(essentially negative when lies between 90° and 270°. 

If PQ were of infinite length, the motion of Q would be 
equal to that of the point E ; but as PQ is of finite length 
(usually from four to eight times AP), Q is drawn toward A 
through the distance PQ - EQ = I - ^P - P? sin^ 



So 



MOTION BY LINKWORK. 



195 



that when AP has moved to its mid-position Aj} or Ap' (or, 
as it is frequently expressed, -4P is on the half-centre), Q 
will have passed its mid-position M by the distance qM = 
AM - Aq = I - sjl" - E\ Also when Q is at M, P will 
be at some point >S' or S' (Fig. 117), intermediate between 




Candp or j/. These points may be readily determined, for 
in this case AQ = l\ hence AQS and AQS^ are equal isos- 
celes triangles, and cos = — — = — . The velocity ratio 

A\^ zl 

of P and Q varies for each instant, but may be determined 
at any time by means of the instantaneous centre (Art. 25) 
or by resolving the velocities, as in Fig. 118. 




Let . V be the linear velocity of P, and v that of Q. 
■Resolve these along and perpendicular to the link PQ ; then, 
as shown in Art. 24, the components along the link must 



196 ELEMENTARY MECHANISM. 

be equal ; that is, 

V cos aPc = V cos bQf. (i) 

Draw AN perpendicular to AQ, and intersecting the link 
(produced) at JSf; draw Ad perpendicular to PQ. Then 
angle a Pc = angle PAd; and angle bQf = angle NAd. 
Hence 

cos aPc = cos PAd = :^^, (2) 

and 

cos 6Q/= cos NAd = — . (3) 

Substituting these values in equation (1), we get 

Hence, 



^^zl=^xzl- w 



a variable quantity. It is evident from this expression that 
when AN = P, the velocities of P and Q are the same. This 
will occur when AP is perpendicular to AQ, as at Ap, Ap' 
(f'ig. 116), in which case AP coincides with AN\ and it 
will also occur when AP occupies such a position that the 
triangle APN is isosceles. To determine the angle 6 which 
will give this position of AP^ we have, from similar triangles, 
AN'.PE::Aq:EQ. 



R''sm^e-\- Rco^O' 



AN= P = PE x^ = Psm ef^ , 
From this equation we deduce 



sme^j^^islsWri'-i)' 



179. The distance through which the point Q is drawn 
toward A by reason of the finite length of the link (Art. 
178) increases rapidly as the link becomes shorter. If we 



Motion by lixkavork. 



19T 



inake the liuk of the same leugtli as the crank arm, as in Fig. 
119, the point K (Fig. 116) coincides with A^ and the path 




T^ig. 119 



of Q is AL = 2E, as before. But Q is drawn toward AP 
so rapidly on account of the angularity of FQ, that when 




Fis.i30 



AP is perpendicular to AL, Q coincides with A and has 
completed its stroke. If we produce QP to F, making 



198 



ELEMENTARY MECHANISM. 



PV = PQ = AP (Fig. 120), it is evident that as AP 
revolves as indicated by the arrow, Q will move from L to 
A^ and V will move from A to W. If now we continue the 
motion of AP^ Q will be driven past A to M^ and V will 
return to A. Thus, the revolution of AP will cause Q to 
move over the path LM, and V over the path WX. By this 
means, the arm AP can be made to move the two ends of a 
link of twice its length through paths at right angles, and 
each equal in length to 4:AP. 

180. We have thus far considered the end of the link, Q, 
to travel in a path of which the direction passes through the 
axis A, but this path may be a straight liue not passing 




through A, as in Fig. 121. In this figure, let AP be a crank 
revolving about the centre A and connected by the link PQ 
to the point Q travelling in the straight line KL. An arc of 
a circle struck about centre A, with radius I — P, will cut 
the line KL at /f, the end of the stroke ; and the inivard 
dead point D will lie in the straight line KAD. Similarly, 
the other end of the stroke, i, and the outward dead point 
O, may be found by striking an arc about centre A^ with 
radius AL — I -\- P. 

The position of the point Q corresponding to any given 
position of AP may be thus determined : — 



MOTION BY LINKWORK. 199 

Let the perpendicular distance between A and the line KL 
be AM = e, and let the angle PA3I = 0. Then 

3iq = MS^SQ, MS=EsmO; 



SQ = sjp - Fs' = sjl' - (e ± B cos Oy. 
Hence we have 



MQ = EsinO + \/l^ - (e ± E cos Oy. 
Also 



ML = \J{1 + Ey - e\ 



From these expressions, the distances QL and QK can be 
determined. 

181. By comparing Figs. 116 and 121, it will be seen, 
that, in the former, the outward and inward dead points, C 
and D, lie in one straight line ; while in the latter, they 
depart from a straight line by the angle DAV = KAL. 

This angle increases as A3f is increased, and as the ratio — 

is increased. The practical result is, that, supposing AF to 
revolve uniformly in the direction of the arrow, the point Q 
will move over its path from L to K in less time than from 
K to X, -the times being proportional to the arcs DGC and 
DFC. 

182. Eccentric. — A particular form of this class of 
link motions is the eccentric, shown in Fig. 122. A circular 
disc called the eccentric, or the eccentric sheave, has its centre 
at the point P, and is made sufficiently large to embrace the 
shaft at A, to which it is fastened. The eccentric is enclosed 
by a strajJ or bcind, FG, in which it revolves. This strap is 
rigidly connected to the rod or bar UN, by which motion 
is transmitted to the point Q. It will be seen, that, as the 
eccentric turns about A and slides within the strap, it will 
communicate exactly the same motion to the point Q as 
would be given by a crank arm AF and link FQ. In fact, 



200 ELEMENTARY MECHANISM. 

it is used as a substitute for small cranks on account of 
the practical difficulties in the formation of the latter. The 
travel of the point Q will, as in Fig. 116, be equal to 2AP, 




The term, throw of an eccentric, is given, by various 
authorities, either to the arm AP^ or to twice that distance ; 
and hence the meaning of the term is often ambiguous. 



Class II. 

Transformation of Continuous Rotation into Rotative Reciprocation^ 
and the Reverse. 

183. In Fig. 123, let J.P and BQ be arms turning about 
the fixed centres A and B respectively, and connected by the 
link PQ. If AP be rotated about A^ it will compel BQ to 
oscillate between the positions Be and 5e, or Be' and Be'^ 
according as the arm BQ has been previously placed above 
or below the centre B. 

Let AP = B,BQ = r, 
PQ = I, AB == d. 

To find the dead points : About -4 as a centre describe 
circular arcs with radii I -\- R and I — JR. They will cut the 
circle about centre B^ radius r, in the points c, c', and e, e', 



MOTION BY LTNKWORK. 



m 



i'espectivel3\ Those give tlie outward find inward dead 
points for B, and hence the luiiits of the oscilUition of r. 
Drawing the pieces in these extreme positions, it will be seen 
that we obtain a series of triangles, of which the base is 
always the line of centres AB ( = cl), and of which the other 
two sides are r, and I + B oy I — R. We will term these 




inig. 1^3 



dead-point triangles. As long as we can construct such tri- 
angles with a sensible altitude, it is clear that there can be 
no dead points for r, and hence the rotation of R will cause 
r to oscillate. But if with any assumed values of i?, /, and 
r, the triangle will reduce to a straight line, r will have dead 
points, and we can no longer control the direction of its 
motion by the single combination shown. Thus, in order 
that the rotation of an arm R may produce oscillation of an 
arm ?•, we must have r greater than i?, and also 

d -[- r>l + R. 

d - r<l - R. 



These conditions are fulfilled by the proportions employed 



in Fig. 123. 



202 



ELEMENTARY MECHANISM. 



184. Figs. 124, 125, and 126 are inserted to show the 
effects of shortening the distance between centres, retaining 
the same lengths of E, Z, and r, as in Fig. 123. 



d=l+r-R 




Fig. 1S4 



In Fig. 124, d = I -i- r ~ E, and we have an outward 
dead point for r, simultaneously with an inward dead point 
for E. In Fig. 125, d = I -\- E — r, and we have an in- 
ward dead point for ?-, simultaneously with an outward dead 



4--=l+R 




IFig. 1S5 



point for E. These two arrangements are therefore faulty ; 
since when r is at a dead point, we cannot control the direc- 
tion of its motion, as it is free to move either up or down 



MOTION BY LINKWORK. 



203 



from that point. In Fig. 126, d is made still shorter, and it 
will be seen that r has dead points, G and G\ when M is at 
Ag or Ag' respectively. Hence it cannot perform complete 
revolutions and drive r. 




IPig, 1S6 



As M moves in either direction over the arc gEE'g', r may 
move in either direction over the path G'e'G'G or GeGG' , 

185. Problem. — Given (Fig. 127) the distance AB 
between two fixed centres A and J5, to find the lengths of 




Fig. 137 



arms and links which will cause the following arm BQ io 
oscillate between the positions jBQ and BQ[ when the driving 
arm AP makes continuous rotations. Assume any convenient 
length for the arm ^Q, and strike an arc with that length as 



204 ELEMENTARY MECHANISM. 

a radius about B^ thus fixing the points Q and Q. With A 
as a centre, draw the circular arcs Qq and Qc/. Then we 
will have AQ = Aq = I — R\ and AQ' =:= Aq' = I -\- E ; 
.'. qq^ = 2E. Bisect qq' at a, then aq or aq' is the length of 
the arm AP to fulfil the conditions, and Aa is the length 
of the link FQ = I. If these results do not give convenient 
lengths for R and ?, a longer or a shorter length should be 
taken for BQ, 

186. The velocity ratio in this class of linkwork is to be 
determined, as in Arts. 24 and 25, by the ratio of the perpen- 
diculars let fall from the fixed centre upon the line of the 
link. When there are dead points, one or both of the per- 
pendiculars disappear at the instant of passing these points ; 
and this is just as it should be, for no motion is transmitted 
at that instant. 

When the shorter and rotating arm is the driver, its dead 
points occur, as shown in the figure, at the ends of the oscil- 
lations of the following arm, and this arrangement will work 
satisfactorily when the conditions of Art. 184 are complied 
with. In case the oscillating arm drive, however, the dead 
points of the follower must be overcome by the momentum 
of the rotating pieces, increased if necessary by the addition 
of a fly-wheel. 

Class III. 

Transmission of Continuous BotatioUo 

187. Drag Link. — In Figs. 128 and 129, let AP, BQ, 
be two arms turning about fixed centres A and B, and con- 
nected by the link PQ, as before. In order that the continu- 
ous rotation of one may produce a continuous rotation of the 
other, it is necessary that there shall be no dead i^oints. If 
the link PQ {— I) is made equal to Cc, it is evident that we 
will have an outward dead point of i? at (7 with an inward 
dead point of r at c. Or, if I = Ce, we will have simultane- 



MOTION BY LINKWORK. 



205 



ous inward dead points at C and e. Therefore, in order that 
there may be no dead points, we must make I > Cc and < Ce ; 




that is to say, I > r — B + d, and I < 7' -\- H — d; and each 
of the arms H and r must be greater than d. 




Fig. 139 



In this arrangement, the arms may be equal or not ; but in 
either case the velocit}' ratio, being proportional to the per- 
pendiculars from the centres on the link, is constantly vary- 
ing. The arrangement is termed a drag link. As frequently 



206 ELEMENTARY MECHANISM. 

constructed, the arms are of the same length, and the centres 
A and B coincide. 

188. Continuous rotation may be transmitted by making 
the arms i^'and r of Fig. 123 equal, and also I = AB. This 




will give us the arrangement shown in Fig. 130. The arms 
have simultaneous dead points when lying in the direction of 
the line of centres. Hence if R performs complete revolu- 
tions, r may continue to rotate in the same direction, with a 
constant velocity ratio, or it may move from the dead point in 
the opposite direction, with a varying velocity ratio ; so that 
for a given position, AP^ of i^, r may occupy the position 
BQ or Bq. To insure continuous rotation, then, by this 
arrangement, it is necessary to provide some means of com- 
pelling r to continue its motion past the dead points. This 
may be accomplished by one of the following arrangements. 

189. We may connect the a±es by other and similar sys- 
tems, as shown in Figs. 131 and 132. 

When two systems of arms and links, are employed, as in 
Fig. 131, they are generally placed with the arms at right 
angles. 

When three are used, as in Fig. 132, the arms are placed 
at angles of 120°. 

190. Another method consists in the use of a third rotat- 
ing arm, connected to the same link, and so placed that the 
driving arm may lie between the following and the auxiliary 



MOTION BY LINKWORK. 



207 




208 



ELEMENTARY MECHANISM. 



arms. This arm must be equal in length to the other two, 
and must lie parallel to them in all positions. All three 
arms may be located on the same line of centres, as in Fig. 




Fig. 133 



133 ; or the driving arm may be to one side of the line of 
centres of the other two, as in Fig. 134. In the latter case 




JFig. 1-3^ 



VT must be part of or be rigidly connected with PQ, so 
that the points P, F, Q will be the vertices of a rigid tri- 
angle. 

191. Boelim's Coupling-. — Another method, known as 
Boehm's link coupling, is shown in Fig. 135. Two discs 
placed in parallel planes are fixed to parallel shafts, and con- 
nected by two or more links which make an angle with the 
planes of the discs. The distance between the planes of the 
discs must be sufficient to enable the links to pass each other, 



MOTION BY LINKWORK. 



209 



as shown. The velocity ratio in these three arrangements is 
constant. 




Various Applications of Linkwork. 

192. Bell-Cranks. — A form of linkwork known as 
bell-cranks is largely used to change the direction of motion. 
In Fig. 13G let ab be the direction of a reciprocating motion 
which it is desired to change to the direction cd. In the 
angle bTd formed by these directions assume any convenient 
point A ; and from A draw perpendiculars AB and AC on 
ab and cd respectively. If we construct a rigid piece BAC, 
centred at A, we can by means of it produce the change in 
dn-ection desired. This piece is termed a bell-crank, and 
such pieces are largely used in bell-hanging, in the mechan- 
ism of organs, etc. As the angular motion of the arms is 
small, their lengths are sensibly equal to the perpendiculars 
from A upon the lines of action, and hence the velocity ratio 
is sensibly constant. 



210 ELEMENTARY MECHANISM. 

It is clear that we may place the centre A in any one of 
the four angles about T made by the lines of action. If 
placed in the angle bTd or aTc, the direction of motion will 
be as indicated by the arrows ; but if we place it in the angles 
aTd or bTc, the direction of motion along ab being still as 
indicated by the arrow, that along cd will be reversed. 




IPig. 136 



193. In order that the deviation of the points B and 
from the lines of action shall be a minimum, and take place 
on both sides of the line of action instead of wholly on one 
side, the length of the arm AB should be equal to the per- 
pendicular distance AB j^lus one-half the versed sine of the 
angle through which the arm swings on each side of its mid- 
position. This is true in all similar cases, and is illustrated 
by the following example : In a beam engine having a piston 
stroke of six feet, let the distance between the centres A of 
the beam gudgeons, and the centre line BP of the cylinder 
l)e eight feet. Required, the best length for the beam arm. 
In Fig. 137 let AB = distance between centres = 8^ Now 
B is to bisect the length CD, which is the sum of the devia- 
tions of the point E on both sides of BP ; hence BC = BD. 
EC must equal the half stroke, or 3'. 



MOTION BY LIXKWORK. 211 

In the right triangle EC A, we have 



EA^ = EC^ + CA' = ad' 



or (8 + BCy = 9 + (8 - BCy. Solving this equation 
for BC, we find BC = ^ = -r' = 3f\ 



Fig. 137 

Therefore the length of the beam arm AE should be 8' 3|^^ 
in order that the connection E may deviate the smallest 
amount from the line of centres BP. 

194. To Multiply Oscillations l>y Linkwork. — In 

Fig. 138, let AP be an arm rotating about A, and connected 
by a link PC to an arm BC oscillating about B. For one 
revolution of AP, BC will move from the position BC to Be 
and back again to BC. Now connect (7 by a link CE to an 
arm DE oscillating about D. Let ^Pturn from P to p, and 
move BC from C to c, then DE will be moved from E to E' 
and back again to E. Therefore during a complete revolu- 
tion of AP, DE will perform tv:o complete cj^cles of motion ; 
going from E to E^ and back again to E during each half 
revolution of AP. If we connect DE to another oscillating 
arm KH in a similar manner, KH will perform two complete 
cycles for one of DE, and hence four complete cycles for one 
revolution of AP, 



212 



ELEMENTARY MECHANISM. 



The length of the arc of osoinatiou of each arm depends 
on the length of the preceding arm, and on the versed sine 
of the angle through which the latter swings on each side of 
its mid- position. 

C 



Fig. 138 



-^ 


V-- 




/\^ 




> \ 


f 


\ 


/ ^1 


\ 


/ ' 


p \ 


r\ 








V ' 


J 



-J \? 



FV~- 



195. To Produce a Rapidly Varying Velocity from 
Uniform Motion. — In Fig. 139 let AP be an arm oscil- 

B 




Q(Z r 



lating with uniform velocity about A^ and connected by a 
link PQ to an arm oscillating about a centre -S, placed so 



MOTION BY LINK WORK. 213 

that when P is at a dead poiut, BQ may be perpendicular to 
PQ. Starting from the dead point P, the uniform motion 
of ^P produces very little motion of BQ at first, but as P 
moves over the equal arcs Pj), jj>|/, etc., Q will move through 
arcs rapidly increasing in length, such as Qq^ qq\ etc. That 
is, the uniform velocity of P produces a rapidl}^ accelerated 
velocity of Q. When P moves in the other direction, from 
X>'" toward P, the velocity of Q will be rapidly retarded. 

196. Slow Advance and Quick Return by Link- 
work. — In Fig. 140, let AP be a rotating arm, from whose 




uniform motion we wish to derive the motion of a second 
arm, such that its period of advance shall be equal, say, to 
twice its period of return. On a circle about A^ lay off the 
arc PCp — one-third of the circumference; i.e., angle 
PAp =120 degrees ; then AP and Ap must be the posi- 
tions of the driving arm when the following arm is at the 
ends of its stroke, and the driver is at its dead points. 

On PA and Ap produced, lay off PQ = pq =i proposed 
length of link. Then Qq must be the chord of the arc of 
oscillation of BQ^ and the centre B must lie in the perpen- 
dicular bisecting this chord. BQ may be of any length to 



214 



ELEMENTARY MECHANISM. 



give the angle of oscillation QBq desired, consistent with the 
deductions of Art. 184. 

197. Hooke's Coupling- or Universal Joint is a con- 
trivance, belonging to the general class of linkwork, for 
connecting shafts whose axes intersect. 

In Fig. 141, A and B are the two shafts having semi- 
circular jaws at their ends. The connecting and rigid cross 




QPpq is formed with its four arms at right angles and in the 
same plane. The centre of the cross is at 0, the intersec- 
tion of the axes. All four arms are of the same length, and 
turn in bearings at P, p, Q, and q. Let the shaft A be the 
driver, then the ends of the arms, P and p, move in a circle 
whose plane is perpendicular to the axis of A ; and the ends, 
Q and g, move in a circle whose plane is perpendicular to 
the axis of B, We will term these arms the driving and fol- 
lowing arms respectively. The planes of the circles evidently 
intersect in a line through 0, perpendicular to the plane of 
the axes, and the angle between the planes is equal to the 
angle between the axes = p. 

198. Let a plane through 0, and perpendicular to the 
driving axis, be taken as the plane of projection. Then, 
Fig. 142, the circle described with radius OP = Op about 
as a centre, will represent the path of the points P and p. 
Let the plane of the axes A and B be perpendicular to the 
paper, intersecting it in OE. Then COD^ perpendicular to 
OE^ will be the intersection of the planes of rotation of the 



MOTION BY LTXKWORK. 



215 



driving and following arms. Draw the radius OF^ making 
the angle EOF = /3 = the acute angle between the axes ; 
and from F draw a line parallel to COD, and intersecting 
OE ii_ K. Then 



OK 
OF 



OK n 

^ = cosA 



and an ellipse constructed with CD as its major and OK as 
its semi-minor axis will represent the projection, on a plane 
perpendicular to the driving axis, of the path of the following 
arms. 




Fig. 143 



Suppose a driving arm to move from the position OC to 
OP through the angle COP = 0. Then a line OQ drawn 
perpendicular to OP will be the projection of a follow^ing 
arm which has moved from OK, while OP moved through 
the angle COP = EOQ = 0. OQ is perpendicular to OP, 
since the latter lies in the plane of projection ; and hence the 
angle POQ is shown in its true size. The point Q has moved 
through the actual vertical distance Qn, although the actual 



216 ELEMENTARY MECHANISM. 

path of Q is a circle of radius equal to OE. Therefore, if 
through Q we draw a line parallel to OE, and connect OR^ 
then EOR will be the actual angle through which the follow- 
ing arm has moved, while the driving arm has moved through 
COP = EOq = e. Let angle EOR = cf>. 

SQ OK o 

— — = — = cos p. 
SR OE ^ 



tan <^ _ mR 
tan Om 


. nQ _ On 
On Om 


Hence 




(1) 


tan(^ = tan 



To obtain the velocity ratio, we must differentiate this ex- 
pression, whence 

^ ^ cie cos^e 1 + tan^c^ a 

Eliminating (^ and 6 in turn from Equation (2) by means of 
Equation (1), we get 

(Q\ ct' _ cos/3 



a 1 — sin^^sin^yS 

(A\ £L — 1 — cos^ <i> sin^ y8 

^ ^ a ~ cos/3 * 

Starting with a driving arm at 00 and a following arm at 
OK, we measure the angles and <^ from these positions re- 
spectively. 

199. The exp;ressions (3) and (4) will have minimum 

values when sin ^ = 0, and cos (/> = 1 ; in that case — = cos /3, 



a 



and and (j> both = 0, tt, 2 tt, etc. That is, the minimum 
values of the velocity ratio occur when a driving arm is at 
OC or Oi>, and the following arm is at O^or KO produced. 



MOTION BY LINK WORK. 217 

Maximum values occur when sin 6=1', cos ^ = 0. In 

f -I q 

this case — = , and and of) both = — , — , etc. That 

a cos/3 2 2 

is, the velocity ratio has its maximum value when the driving 

arm is at OE or EO produced, and the following arm is at 

OD or 0(7. 

Hence we see that during each revolution there are two 

maximum and two minimum values of the velocity ratio, and 

that it varies between the values and cos B. 

cos^ ^ 

Between the maximum and minimum points there are four 
points where the ratio is unity. 

The variation in the velocity ratio increases as the angle 
between the axes increases, and this fact must, in general, 
govern the employment of this joint. If the variation due 
to the angle between the shafts is not too great for the case 
under consideration, it may be employed ; otherwise some 
other mode of connection must be used. 

200. Double Hooke's Joint. — The variation in the 
velocity ratio may be entirely eliminated by the use of two 
Hooke's joints, arranged as in Fig. 143. 





JFig. 143 



If we construct the connecting piece with the forks at its 
ends in the same plane, and place it so that it makes the 
same angle (^) with both shafts A and 5, the variation at 
one end of the connecting piece will be counterbalanced by 
the variation at its other end ; and thus a uniform motion 
may be transmitted from A to B. For, in Fig. 143, let the 
plane of the axes be the plane of the paper, then considering 



218 ELEMENTARY MECHANISM. 

A as the driver, the velocity ratio between A and ab is at its 
maximum ; i.e., 



coS)8 



Also, considering ab as the driver, the velocity ratio between 
ab and B is at its minimum ; i.e., 

a rt 

- = cos/3. 
Multiplying these two equations together, we get 



^=1. 



It will be clear upon examination that the variations thus 
balance each other throughout the revolution, so that — is 

a 

always unity ; in other words, the velocities of the two prin- 
cipal axes are always equal. If the cross ends of ab are in 
planes perpendicular to each other, the variations, instead of 
neutralizing each other, will evidently act together, and make 
the total variation in velocity of the two principal axes 
greater than if one joint only were employed. 

Intermittent Linkwork. 

201. Click and Katcliet. — An example of intermittent 
linkwork is the dick and ratchet-ivheel, a simple form of which 
is shown in Fig. 144. An arm AB oscillating about A has 
jointed to it at ^ a dick or catdi BC. Turning about i> is a 
ratdiet-wheel Cc, having teeth generally of the shape shown. 
When the arm AB moves as indicated by the arrow, the end 
C of the click presses against the straight side of a tooth, 



MOTION BY LINKWORK. 



219 



and thus moves the wheel. Centred at the fixed point h is 
a paid or detent^ be, which either rests on the teeth by its 
weight, or is pressed against them by a spring. This serves, 
as shown, to hold the wheel, and prevent its backward motion 
during the back stroke of AB, but offers little or no resist- 
ance during the forward motion of AB. The pawl and tooth 
act upon each other by sliding contact ; and the direction of 
the line of action, or of the pressure between them, is a nor- 
mal to the straight face of the tooth and end of the pawl. 




Let eg be this normal, and let fall upon it the perpendiculars 
bg and De. Then, if the wheel tend to turn backwards, that 
is, in the direction eg, the pawl will tend to turn about b in 
the same direction, eg, or towards the wheel. That is, the 
tendency is to force the pawl and the tooth into closer con- 
tact, which is as it should be. But if the shape of the face 
of tooth and pawl is such that the normal occupies some 
position be^^ond 6, as c7i, the tendency is to turn the pawl 
outward, or to cause it to slide off the tooth. Hence, with 
a straight pawl or click, as shown in this figure, the normal 
to the face of the tooth should pass between the centres 



220 



ELEMENTARY MECHANISM. 



of the wheel and of the pawl. And, by similar reasoning, 
in the cases in which we have a hooked pawl, as in Fig. 148, 
the normal should pass outside of or beyond the centre of 
motion of the pawl. 

202. Reversible Click. — In feed motions, such as 
those of shapers and planers, it is frequently desirable to 
employ a click and ratchet-wheel which will drive in either 
direction. An arrangement similar to that shown in Fig. 145 




Fig. 145 



is then used. In the position shown, the wheel is being 
driven in the direction of the arrow. The teeth, as well as 
the click, are made alike on both sides ; so that when the 
click is thrown over on the other side of the arm AB, as 
shown in dotted lines, the wheel will be driven in the contrary 
direction. 

203. It is usually more convenient to have the driving 
arm and the wheel concentric, as in Fig. 145, rather than as 
in Fig. 144. It is clear, that, in this case, to have an effec- 
tive arrangement, the driving arm should move through such 
an angle that the end of the click shall travel through an arc 



MOTION BY LIXKWORK. 



221 



sliglitl}' greater than some multiple of the pitch arc ; the 
excess being simply to insure that the click shall clear 
the correct number of teeth on its return stroke, and have 
the smallest possible amount of lost motion. Thus, if the 
click and arm drive the wheel ahead two teeth at each for- 
ward stroke, the arc of motion of the click should be a trifle 
over twice the pitch arc, to insure the same amount of motion 
by each forward stroke. 

204. Silent Click. — This contrivance (Fig. 146) avoids 
the clicking noise and the consequent wear of a common 
click. BC is the click, which, in this figure, is made to push 
the teeth. 



Fig. 146 




It is carried l)y one arm, AB^ of a bell-crank lever, which 
has the same centre of motion, ^4, as the ratchet-wheel. The 
other arm of the lever has two studs, E and E'. Between 
these pins is the driving arm AF^ also centred at A^ and con- 
nected by a link GH to the click. The motion of this arm 
in the direction of the arrow drives the wheel in the same 
direction. When the motion of the arm is reversed, it at 
first moves back against E' before it can move the bell-crank 
lever ; and, during this motion, the link GH lifts the click 



222 



ELEMENTARY MECHANISM. 



clear of tbe teeth. Then, by pressing against the pin E\ it 
moves the bell- crank back to the position shown in dotted 
lines. The driving arm then moves ahead again against the 
pin E^ pulling the click into gear with the teeth, as shown ; 
and then, by means of the pin £", drives the wheel ahead, as 
before. 

205. Double-acting- Click. — This arrangement, shown 
in Figs. 147 and 148, may be used when it is desired to drive 
the wheel ahead during both strokes of the driving arm. 



Fig. i^^i^ 




To accomplish this result, the driving arm in Fig. 147 car- 
ries two pusJiing clicks ; and that in Fig. 148, two pi(?Z«i^ 
clicks. The former is the stronger arrangement, and is 
therefore used wherever great strength is required, as in 
ships' windlasses. The centre G being located, the arms 
GK and GH are made equal, and so placed that in their mid- 
positions they will be perpendicular to the lines of action of 
the respective clicks. With these arrangements, two or more 



MOTION BY IJNK^VORK. 223 

detents or pawls are used, so placed that they will prevent 
the ratchet-wheel from turning back more than one-third or 
one-half the pitch. 




V. 



A T-ig. 148 



206. Frictional Catcb. — This contrivance is a sort of 
intermittent link work, founded on the dynamical principle, 
that two surfaces will not slide on each other so long as the 
angle which the direction of the pressure between them 
makes with their common normal at the point of contact is 
less than a certain angle, called the angle of repose. This 
angle depends on the material of which the surfaces are com- 
posed, their condition as to smoothness, and on the lubrica- 
tion employed. For metallic surfaces, moderately smooth, not 
lubricated, the sine of this angle is somewhat greater than 
one-seventh. In Fig. 149, the shaft and rim of the wheel 
to be acted upon are shown in section. AK is the catch 
arm, having a rocking motion about the axis^ of the wheel ; 
the link by which it is driven is supposed to be jointed to 
it at K. K'K" represents the stroke, or arc of motion, of 
the point K\ so that K' AK" is the angular stroke of the 
catch arm. i is a socket, capable of sliding up and down 
on the catch arm to a small extent ; a shoulder for limiting 
the extent of that sliding is shown by dotted lines. The 
socket and the part of the arm on which it slides should be 



224 



ELEMENTARY MECHANISM. 



square, and not round, to prevent the socket from turning. 
From the side of the socket there projects a pin at Z), from 
which the catch DGH hangs. Jf is a spring pressing 

G and H are two 



against the forward side of the catch. 




studs on the catch, which grip and carry forward the rim 
BBCC of the wheel during the forward stroke, by means of 
friction, but let it go during the return stroke. 

A similar frictional catch, not shown, hanging from a 
socket on a fixed instead of a movable arm, serves for a 



MOTION BY LIXKWORK. 



225 



detent, to hold the wheel still during the return stroke of the 
movable arm. 

207. The following is the graphic construction for deter- 
mining the proper position of the studs G and H: — JNIultiply 
the radii of the outer and inner surfaces, BB and CC\ of the 
rim of the wheel, by a coefficient a little less than the sine of 
the angle of repose, — say one-seventh, — and with the lengths 
so found, as radii, describe two circular arcs about A ; the 
greater (marked E) lying in the direction of forward motion, 
and the less (marked F) in the contrary direction. From i), 
the centre of the pin, draw DE and DF tangent to these 
arcs. Then (x, where DE cuts BB^ and //, where DF cuts 
(70, will be the proper positions for the points of contact of 
the two studs with the rim of the wheel. 

The stiffness of the spring ought to be sufficient to bring 
the catch quickly into the holding position at the end of each 
return stroke. 

The length of stroke of a frictional catch is arbitrary, and 
need not be an aliquot part of the circumference of the 
wheel, as is the case with the click motions described. 

208. Another form of frictional catch is shown in Fig. 
150. 




Fig. 15 O 



An arm AB^ centred at (7, rides on a saddle which slides 
on the rim NN of the wheel. A piece EE is attached to 



226 ELEMENTARY MECHANISM. 

one end of the arm, and admits of being pressed firmly 
against the inside of the rim JSfJSf. When the end B is moved 
as indicated, the rim TViV will be firmly grasped or nipped 
between the saddle and the piece EE, and will be forced to 
move to the right. When B is pushed back, a stop prevents 
BCA from turning more than is sufficient to loosen the hold 
of EE^ and the saddle slides freely on the rim. A screw E 
may be employed to bring up a stop H towards the arm 
ACB, and so to prevent the arm from twisting into the posi- 
tion which gives rise to the grip of EE. No motion will 
then be imparted to the wheel, a result which is obtained in 
any ordinary ratchet-wheel by throwing the click off the 
teeth. 



MOTION BY WRAPPING CONNECTORS. 227 



CHAPTER XL 

COMMUNICATION OF MOTION BY WRAPPING CONNECTORS. 

VELOCITr RATIO CONSTANT. 

DIRECTIONAL RELATION CONSTANT. 

Forms of Connectors and Pulleys. — Guide Pulleys. — Twisted Belts. 
Length of Belts. 

209.. It follows from Art. 27, that when the du-ection of 
the wrapping connector of two curves revolving in the same 





Fig. 151 S^ig. 15 S 



plane cuts the line of centres in a fixed point, the velocity 
ratio must be constant. The only curves used in practice 
are circles, the surfaces being surfaces of revolution rotating 
about fixed axes. In order that the motion may be continu- 



228 



ELEMENTARY MECHANISM. 



ous, the ends of the wrapping connector are fastened to- 
gether, forming an endless band which embraces a portion 
of the circumference of each wheel, ov pulley as it is usually 
termed. 

Where a direct or oj^e?! band is used, as in Fig. 151, the 
direction of rotation of driver and follower is the same ; but 
when the band is crossed, as in Fig. 152, the rotations take 
place in opposite directions. 

210. Forms of Connectors and Pulleys. — Various 
materials are in use for wrapping connectors, the material 
depending to a certain extent upon the character and location 
of the machinery. The form of the pulleys depends largely 
upon the material of the connector. For very light machinery, 
such as sewing machines, the hands are usually round, and 
are made of leather, catgut, or woven cord. The pulleys 



ir-ig. 153 



Fig. lS4t 



used with such bands are grooved as illustrated by Fig. 153, 
the band running in this groove. For other machinery, 
where the distance between driver and follower is not very 
great, flat belts are used together with smooth pulleys. These 
pulleys are true cylinders in some cases, but are usually 
rounded to some extent as illustrated by Fig. 154. The 
amount of this convexity, or increase of radius from edge to 
centre of face, varies, according to different authorities, from 
nothing to one-half inch per foot of width of face of pulley. 
Average practice would seem to authorize one-eighth inch rise 
per foot of width. 



MOTION BY WRAPPING CONNECTORS. 229 

Three or four such pulleys of different diameters are often 
made in one piece, the size of the pulleys increasing regularly 
from end to end. Such an arrangement is called a step2')ecl 
pulley ; and by means of two such pulleys, mounted on 
parallel shafts, and placed so that the smallest diameter of 
each is opposite the largest diameter of the other, motion can 
be transmitted between the shafts with a definite number of 
different velocity ratios. The various diameters must be so 
adjusted that the same length of belt (Art. 218) can be used 
in each case, the variation in the velocity ratio being obtained 
by simply transferring the belt from one pair of pulleys to 
another. 

Flat belts are generally made of leather, either of a single 
thickness, or of two or more thicknesses sewed, riveted, or 
cemented together. The grain or hair side should be placed 
next the pulley. Woven cotton covered with vulcanized 
India rubber, and known as " rubber belting," is also largely 
used, particularly where dampness renders leather unfit. 
Paper and sheet iron have also been used to some extent. 




For transmitting power over long distances, wire rope is 
used. The rims of the pulleys are grooved as shown in Fig. 
155, the bottoms of the grooves being filled with wood, 
leather, oakum, or some other material, to reduce the wear 
of the wire rope. 



230 



ELEMENTARY MECHANISM. 



For the running rigging of ships, tackles, and hoisting 
machinery, hemp or other similar rope is used with smooth 
grooved pulleys similar to the one shown in Fig. 153. 




E"ig. 158 
Where great strength is required in small compass, iron 
chains are used. The links of the cliains are of various 
shapes. The pulleys are formed to fit the links more or less 
nearly, or with teeth to enter the links, and thus prevent 
slipping. Figs. 156, 157, and 158 illustrate forms of chains 
and pulleys. 



MOTION BY WRAPPING CONNECTORS. 231 

211. Tigliteiiiiig- Pulleys. — When smooth pulleys are 
used, the motion is transmitted directly by the friction be- 
tween the belts or bands and the pulleys. Ordinarily the 
tension of a belt, properly fitted, is sufficient to produce the 
necessary adhesion. But in some cases, tightening pulleys 
are employed to prevent slipping ; as, for example, they are 
frequently employed for this purpose on the driving belts 
of stationary steam-engines. These , tightening pulleys are 
pressed against the belt by weights ox ^ springs, and thus 
maintain a constant tension, or are mounted in a frame which 
can be adjusted in position by screws. 

212. Shifting Belts. —A flat belt may be easily shifted 
from one position on a cylindrical pulley to another position 




by pressing the belt in the required direction on the advancing 
side, while pressure on the retreating side will produce no 
effect. Thus, if we press a belt in the above manner as 
shown in Fig. 159, it is clear, that, as the pulley continues 
to revolve, the successive portions of the belt come into 
contact with the pulley at points to the left of the original 
position, and as the revolution of the pulley carries them in 
a direction perpendicular to the axis, the position of the 
belt on the pulley is gradually changed. If we had pressed 
the belt on that part which had left the pulley, its position 
on the pulley would not have been affected. 

From the above it follows, that, if the central line of the 



232 



ELEMENTARY MECHANISM. 



advancing side of a flat uniform belt is kept in the central 
plane of the pulley, it will run true without any tendency to 
leave the pulley. 

213. Convexity of Pulley If we place a flat belt on 

a convex pulley, as shown in Fig. 160, the tension at the 




edge D will evidently be greater than at the edge F, Con- 
sequently the tendency will be to throw the belt into the 
position shown dotted. If we now rotate the pulley, the 
belt will, as shown in the preceding article, be moved to 
the left or towards the largest diameter of the pulley. It is 
for this reason that the convexity is given to pulleys, so that 
if for any reason the belt commences to come off, the in- 
creased tension of one part will bring it back to a central 
position. 

214. Twisted Belt. — In Fig. 161, let ^ be a fixed axis 
carrying the pulley Z>, and let B be an axis carrying the 
pulley E. At first consider the axis B and pulley E to 
occupy the position shown dotted, so that A and B are 
parallel, and D and E are in the same plane. Let SS be 
the common tangent to the two pulleys, drawn on the sides 
from which the belt is delivered, and in the central planes 
of the pulleys. Now, let axis B and pulley E be turned 
about SS into some other position such as that shown in full 
lines. Then SS will be the intersection of the central planes 



MOTION BY WRAPPING CONNECTORS. 



233 



of the pulley. If, now, the pulleys be rotated as indicated by 
the arrows, it follows, that since the points a and h are in the 
central plane of E^ and a' and h^ are in the central plane of i), 
the advancing, side of the belt is in each case in the central 
plane of the pulley considered, and the belt will not tend to 
run off (Art. 212). But if the pulleys be rotated in the 
opposite direction, the belt will immediately run off. Hence 




2Fig. 161 



this arrangement can only be used when the axes are always 
to revolve in the same direction. In laying out twist-belt 
motions^ the circles Z) and B should be taken equal to the 
largest dianieter of the respective pulleys plus the thickness 
of the belt. 

215. Guide Pulleys are used to change the direction 
of belts. In Fig. 162, let ca be the direction of a belt which 
we wish to change to the direction ah. If we place a fixed 
pulley of any convenient diameter in the angle cab with its 
axis on the line bisecting this angle, and so that the lines 
ca and ah are both tangents to the pulley, it is evident that 



234 



ELEMENTARY MECHANISM. 



by means of this pulley the desired change may be effected. 
If the directions do not intersect within a convenient dis- 





T^ig. 163 Fig. 163 

tance, as in Fig. 163, connect them by a line de at any con- 
venient points, and place guide pulleys in the angles at d and 
e, as shown. 

216. By means of guide pulleys we can connect axes 
neither parallel nor meeting in direction, so that they may 




Fig. 164 

be rotated in either direction. In Fig. 164, SS is the inter- 
section of the central planes of the pulleys A and B. Assume 



MOTION BY WRAPPING CONNECTORS. 



235 



points c and c' in this line, and draw tangents to the pulleys 
A and B. Then the guide pulleys C and C should evidently 
be placed in the planes of these tangents, and so as to be 
tangent to ca, c&, and cci! ^ c'b\ respectively. By this 
arrangement the direction of the belt where it leaves a 
pulley is always in the central plane of the next pulley, and 
hence the belt can be run in either direction without tending 
to leave the pulleys. 

217. In Figs. 1G5 and 16G are shown applications of 
guide pulleys, the rotation being always in the same direc- 
tion. In Fig. 1C5, two axes which lie in the same plane and 
make a small angle with each other are connected, so as to 




Fig. 165 



be capable of rotation in the direction of the arrow. One 
guide pulley is used, and the arrangement depends on the 
principles of Art. 214. 

In Fig. 166, two pulleys on parallel shafts, but not in the 
same plane, are connected, so as to be capable of rotation 
in one direction by means of two guide pulleys fixed on the 
same shaft. The diameter of the guide pulleys should be 



236 



ELEMENTARY MECHANISM. 



equal to the distance between the central planes of the 
driving and following pulleys. 




*218. Length of Belt ^he actual length of belt re- 
quired in every case is best determined by actual measure- 

L 




r-ig. 167 



ment over the pulleys, or by measurement on a scale drawing. 
It may, however, be calculated in the following manner. 
In Figs. 167 and 168, let J. and B be the axes of two pulleys 



MOTION BY WRAPPING CONNECTORS. 



287 



connected by belts, KP and Iqj being the straight parts of 
the belt. Draw liAH and gBG perpendicular to AB; AK 
and Ak^ BP and Bp^ radii of the pulleys to the points of 



H K 




Fig.aes 



tangency of the belt ; and BL parallel to KP. Let HAK = 
PBg = ABL = (^ ; AB = d; AK = R; BP = r. Then, 
for a crossed belt (Fig. 167), 



.EP= Vc^2 _(22 + r)2. 



Arc of contact for pulley, radius R 

= i?(7r + 2cf>) = i^^TT + 2 arc sin ^ "^ ^ Y 

Arc of contact for pulley, radius r 

= rf TT + 2 arc sin ■ — ). 

\ ^ I 

.• . Total length of crossed belt = L 

= 2sJd'-{B + ry +{R + r)(^ + ^ arc sin ^^^ 

For an open belt (Fig. 168), 



KP = sf¥^R - ry. 



238 ELEMENTARY MECHANISM. 

Arc of contact for pulley, radius R 

= i^/^TT + 2 arc sin =^^^'Y 

Arc of contact for pulley, radius r 

I , n-r\ 

— rl TT — 2 arc sin ). 



,' . Total length of ojmi belt = L= 2\/d^-{E-ry 

R-r 



+ Tr{R + t) +2{R ~ r) arc sin 



d 



It is to be noted, that, for a given value of d, the length of 
a crossed belt depends upon the sum of the radii of the 
pulleys, while the length of an open belt depends both upon 
the sum and difference of the radii. It follows from this, 
that one crossed belt can be used to transmit different 
velocity ratios between two shafts, with the single condition 
that the sum of the radii of each pair of pulleys must be 
the same. For example : with any given value of d^ the 
same belt, crossed, will exactly fit pulleys having diameters 
of 4 and 16, 6 and 14, 8 and 12, 10 and 10 ; the value of 
R -\- r in each case being 10. But these pulleys could not 
be exactly fitted with the same length of open belt. 

219. Approximate Formulae. — As the above exact 
formulae are cumbersome, the following approximate equa- 
tions are introduced. All dimensions are best taken in 
incJies, and the signification of the letters is the same as 
above. The formulae will give results which are safe within 
the prescribed limits. 

For crossed belt, L = 3f(i2 + r) + 2d. 

R 4- r 
To be used when — -i— does not exceed 0.23. 



MOTION BY WRAPPING CONNECTORS. 239 

For opeu belt, L = ol{R + r)+ 2d. 

R — r 

To be used when does not exceed 0.16. 



Within these limits the results are a trifle large, while 
beyond them they fall short. 

220. Wrapping connectors may be used to transmit mo- 
tion when the directional relation or the velocity ratio, or 
both, are variable. The result is obtained by using non- 
circular pulleys, or by winding the band in a spiral groove 
of variable radius. 

In such cases the length of the band is not usually con- 
stant, and tightening pulleys must usually be emploj^ed to 
insure the requisite tension. 

In practice, variable conditions are so much better met 
by other modes of connection, that wrapping connectors 
are scarcely ever used for this purpose ; and hence no 
discussion of such use is here given. 



^40 Elementary mechanism. 



CHAPTER XII. 



TRAINS OF MECHANISM. 



Value of a Train. — Directional Belation in Trains. — Clockwork. — 
Notation. — Method of designing Trains. — Approxitnate Num- 
bers for Trains. 

221. The required velocity ratio of two motions being 
given, it is always theoreticaUy possible to obtain this ratio 
by the use of one of the elementally combinations described in 
the previous chapters. It often happens, however, that this 
ratio is so small, or so large, that, practically^ the motion is 
better communicated by a train of such combinations ; each 
piece being at the same time the follower of the piece that 
drives it, and the driver of the piece that follows it. 

For convenience, let us first consider the case in which all 
the pieces are circular wheels revolving about fixed axes. 
The usual arrangement in such cases is to secure two unequal 
wheels upon each axis, except the first and last, and to make 
the larger wheel on each axis gear with the smaller wheel on 
the next axis. 

222. Value of a Train. — Let there be m such axes, 
and let us designate by e the value of the train ; that is, the 
ratio of the angular velocities of the first and last axes, or, 
what amounts to the same thing, the ratio of their syn- 
chronal rotations. 



TRAINS OF MECHANISM. 241 

Let aj, as, ag, . . . a,,^ be the angular velocities of the suc- 
cessive axes. Then we have 



, = ^ = 1^ X 1^ X ^ X . . . ^- (1) 

a, a, ttg ttg a^_i ^ 



That is, the value of the train may be found by multiplying 
together the separate ratios of the angular velocities of the 
successive pairs of axes. 

Again, let the synchronal rotations of the successive axes 
of the train be X^, L^^ ig, . . . L^. Then we have 

' = t = txSxr:x---^- (^) 

That is, the value of the train may be found by multiplying 
together the separate ratios of the synchronal rotations of 
the successive pairs of axes. The value of e is, of course, 
the same in both the above equations. This value will not 
be affected by the substitution, for any of the intermediate 
ratios, of any other two numbers that are in the same propor- 
tion ; hence we may express the values of those ratios in the 
terms that may most easily be obtained from the train whose 
motions we wish to consider. 

Letting a = angular velocity of one of two wheels in gear, 
R its radius, N its number of teeth, P its period, or time of 
one rotation, and L its number of rotations in a given time ; 
and letting a', R\ N\ P', and L' be the corresponding quan- 
tities for the other wheel, we have (Art. 35), 

a R' N' P' i' \^ 

which equation will enable us to write the proper ratio in 
each case. ♦ 



242 ELEMENTARY MECHANISM. 

For instance, let -^j, N^, ^3, . . . iV^_i be the numbers of 
teeth of the drivers on the successive axes, and let n^, n^, n^, 
, . . n^n be the numbers of teeth of the corresponding follow- 
ers. Then we may write, by making the proper substitutions 
for the intermediate ratios in Equation (1), 

o^ L^ iVj iV^ JV3 iV^.i 

aj i>i n^ n^ n^ n^ 



n^ X n^ X n^ X . . » n^ 



('f) 



That is, the value of the train is equal to the quotient 
obtained by dividing the continued product of tlie numbers 
of teeth of all the drivers by the continued product of the 
numbers of teeth of all the followers. 

It is obvious, that, in a train of this kind, the number of 
drivers, as well as the number of followers, is always one 
less than the whole number of axes. 

223. Practical Example. — It is not necessary that all 
the ratios should be expressed in the same terms. As before 
stated, it is simply necessary to use, for each ratio, two num- 
bers in the proper proportion. 

For example, let there be a train of six axes, connected as 
above described. 

Let the first axis revolve once per minute, and let the 
second axis revolve once in fifteen seconds. Hence 

Pi _ 60 

:^-i5* 

Let the second axis revolve three times while the third 
revolves five times. Hence 

i3_ 5 



TRAINS OF MECHANISM. 243 

Lpt the third axis carry a wheel of sixty teeth, driving a 
wheel of twenty-four teeth on the fourth axis. Hence 

^3 _ 60 
n, 24:' 

Let the fourth axis carry a pulley of twenty-four inches 
diameter, driving, by means of a belt, a pulley of twelve 
inches diameter on the fifth axis. Hence 

JR, - 12* 

Let the fifth axis turn with an angular velocity two-thirds 
as great as that of the sixth axis. Hence 

«,_1_3 

Substituting these ratios for the successive terms of Equa- 
tion (1), we get 

ar. P, Xo -^q Ra OLr 

"■1 ^2 -^2 »4 ^S "5 

-^^ x^ x^^ x^^ x^ -50 

That is, the angular velocity of the last axis is fifty times as 
great as that of the first ; in other words, the last axis will 
make fifty revolutions in the same time that the first axis 
revolves once. 

224. Directional Relation in Trains. — Li this man- 
ner, we may find the syncTironal rotations of the extreme 
axes in any train of mechanism. Their directional relation 
depends on the number, and the manner of connection, of 
the axes. In a train consisting solely of spur wheels or pin- 
ions on fixed parallel axes, the direction of rotation of the 
successive axes will be alternately in opposite directions. 



244 ELfiMENTAilY MECHANISM. 

Hence, if the train consists of an odd number of axes, the 
first and last axes will revolve in the same direction ; if it 
consists of an even number of axes, they will revolve in 
opposite directions. 

In this connection, it must be remembered that an annular 
wheel (Art. 36) revolves in the same direction as its pinion. 

When the axes in a train are not parallel, the directional 
relation of the extreme axes can only be ascertained by tra- 
cing the separate directional relations of each successive pair 
of axes in order. 

Two separate wheels in a train may revolve concentrically 
about the same axis ; as, for example, the wheels to which 
are attached the hands of a clock. In this case, one of the 
wheels is fixed on the axis as usual, and the other is fixed on 
a tube, or canyion as it is sometimes called, which revolves 
freely on the first axis. 

If these wheels are to move in opposite directions, a single 
bevel wheel may be used to connect them ; but if they are to 
turn in the same direction, as in a clock, they must be made 
in the form of spur-wheels, and connected by means of two 
other spur-wheels fixed to an axis parallel to the first. 

225. Idle Wheel, — Let a spur-wheel be placed between 
and in gear with two other spur-wheels. Let the radii of 
the first, middle, and last wheels be i^^, R^^ i^g, and let their 
angular velocities be a^, a^^ a^. Then we have, for the first 
and middle wheels, 

and, for the middle and last wheels, 

^ _ ■??. 

Multiplying these equations together, we get 



TRAINS OF MECHANISM. 



245 



That is, the velocity ratio of the two extreme wlieels is pre- 
cisely the same as though they were in immediate contact. 
The intermediate wheel is called an idle wheel; and, though 
it does not affect the velocity ratio, it does affect the direc- 
tional relation. For, if the two extreme wheels were in 
direct contact, they would revolve in opposite directions ; 
but, by the introduction of the idle wheel, they are caused to 
revolve in the same direction. 

226. Clockwork. — A familiar example of the employ- 
ment of a train of wheels is afforded by a common clock. 




In Fig. 1 69 is shown the arrangement of the wheels in a 
clock of the simplest kind. A is the barrel^ and around it is 
w^ound a cord to the end of which is fastened the weight W. 
On the same axis with A is fixed the spur wheel B^ which 
gears with the pinion h on a second axis. On the latter is 



246 ELEMENTARY MECHANISM. 

also fixed the spur wheel (7, gearing with the pinion c on the 
third axis. This axis also carries an escapement wheel D 
(Art. 172), the verge or anchor d being fixed to the fourth 
axis, to which the pendulum is also hung at e. One tooth of 
the escape-wheel crosses the line of centres for every two 
vibrations of the pendulum. Let the time of one vibration 
of the pendulum be t seconds, and let the escape- wheel have 
A teeth ; then the period or time of one complete rotation of 
this wheel is 2^A seconds. To take a simple case, let the 
pendulum be a seconds pendulum ; then ^ = 1, and if A = 
30, the swing-wheel will make one complete revolution in 
2^A = 2 X 30 = 60 seconds = 1 minute. Let B have 48 
teeth ; 6, G teeth ; (7, 45 teeth ; and c, 6 teeth. Then we 
have, for the value of the train connecting the barrel axis 
and the escapement axis, 

a, X, 48 X 45 

bO. 



That is, the escapement axis (or arbor, using the term 
employed by clockmakers) will make sixty revolutions while 
the barrel arbor makes one. Hence the barrel arbor will 
revolve once in sixty minutes, or one hour. The barrel A is 
not permanently secured to this arbor, but is connected to it, 
or to the wheel B, by means of a click and ratchet (Art. 
201); so that, while it is free to move in one direction, its 
rotation in the other direction compels the wheel B to rotate 
with it. This arrangement permits the barrel to be rotated 
so as to wind up the cord without affecting the rest of the 
train. The number of times that the cord is wound round 
the barrel evidently depends on the length of time that the 
clock is to run without being wound. Generally not over 
sixteen coils of cord are so employed, which, in our clock, as 
the barrel arbor revolves once an hour, would be sufficient to 
make the clock run sixteen hours without re-winding. 



TRAINS OF MECHANISM. 247 

227/ The Iraiu of wheel- work just described is solely 
destined for the purpose of commuuicating the action of the 
weight to the pendulum in such a manner as to supply the 
loss of motion from friction and the resistance of the air. 
But besides this, the clock is required to indicate the hours 
and minutes by the rotation of two separate hands, and 
accordingly two other trains of wheel- work are employed for 
this purpose. The train just described is generally contained 
in a frame, consisting of two plates, shown edgewise at kl^ 
mn, which are kept parallel and at the proper distance by 
means of tliree or four pillars not shown in the diagram. 
Opposite boles are drilled in these plates, which receive the 
pivots of the axes or arbors already described. But the axis 
which carries A and B projects through the plate, and other 
wheels E and F are fixed to it. Below this axis, and paral- 
lel to it, a stout pin or stud is fixed to the plate. On this 
stud revolves a tube, to one end of which is fixed the minute- 
hand M, and to the other the wheel e in gear with E. In 
our present clock, the wheel E, being fixed to the barrel 
arbor, revolves once an hour ; and as the minute-hand must 
also revolve once in that period, the wheel E and e must be 
equal. A second and shorter tube is fitted upon the tube of 
the minute-hand so as to revolve freely, and this carries at 
one end the hour-hand H, and at the other a wheel, /, which 
is driven by the pinion F. As / must revolve once in twelve 
hours, it must have twelve times as many teeth as F. 

228. Notation. — In discussing problems concerning 
trains of mechanism, we soon feel the need of some scheme 
of notation, whereby we may show, clearly and concisely, all 
the facts concerning the train which affect the transmission 
of motion. It is desirable to show, primarily, the order and 
nature of the several parts, and the manner in which the 
motion is transmitted ; but such a scheme should also admit 
of the addition of dimensions and nomenclature, and should 
afford a ready means of calculating the velocity ratio. 



248 



ELEMENTARY MECHANISM. 



Let the wheels be represented by their numbers of teeth, 
and write these numbers, beginning with the first driver, in 
horizontal lines ; all the wheels that are on the same axis 
having their numbers written on the same horizontal line, 
and all the wheels that are in gear having the numbers of 
the followers written vertically below those of the respective 
drivers. 

229. Example. — Thus, in the principal train of the 
clock (Fig. 169), if the letters represent the wheels, we 
should write the train thus : — 

B 

or, employing the numbers already selected, 



48 
6 — 45 
6 



30 



Similarly we may represent the whole mechanism of our 
clock, adding to the numbers the names wherever it may be 
thought necessary. Thus — 



Barrel 48 

6 — 45 



25 



6 — 30 swing-wheel. 



25 minute- 
hand. 



48 hour- 
hand. 



TRAINS OF MECHANISM. 249 

The above shows clearly the three trains of mefhanisiii 
from the barrel to the swing- wheel, the minute-hand, and the 
hour-hand respectively. It also distinctly classifies the pieces 
as drivers or followers, as the case may be, and shows the 
nature of their connection ; that is, whether they are per- 
manently fixed to the same axis, or connected by gearing. 
In case other connections are employed, such as links or 
bands, this must be written in the diagram, or expressed by 
a proper sign. 

230. Method of Designing* Trains. — We are now 
ready to undertake the solution of a problem of considerable 
importance in the contrivance of mechanism ; namely, Given 
the velocity ratio of the extreme axes or pieces of a train, to 
determine the number of intermediate axes, and the propor- 
tions of the wheels, or numbers of their teeth. For simpli- 
city, we will suppose the train to consist of toothed wheels 
only ; for a mixed train, consisting of wheels, pulleys, link- 
work, and sliding-pieces, can be calculated upon the same 
principles. Let the synchronal rotations of the first and last 
axes of the train be L^ and L^,, respectively, and let N^^ N.^, 
. . .etc., be the numbers of teeth in the drivers, and n^^ n^, 
etc., the number of teeth in the followers ; then the value of 
the train is 

~ L^~ W.2 X n^ X ?i4 X . . . n^ 

both numerator and d.enominator of this fraction being com- 
posed of m — 1 terms. 

The value of e being given in this shape, an equal fraction 
must be found, whose numerator and denominator shall each 
admit of being divided into m — 1 factors of convenient 
magnitude for the number of teeth of a wheel. 

The value of m, that is, the number of axes, is sometimes 
given with the other data of the problem, but more usually 
it is one of the quantities that are to be determined. 



250 ELEMENTARY MECHANISM. 

The order of succession of the drivers and followers is a 
matter of indifference, so far as the velocity ratio is con- 
cerned ; for the value of the above fraction will evidently 
not vary with any variation in the order of the factors of 
either the numerator or denominator. 

231. Least IS'imiber of Axes The number of axes 

will evidently depend upon the limits between which the 
numbers of teeth are to be allowed to vary. 

For instance, let lo be the greatest number of teeth that 
can be conveniently assigned to a wheel, and let p be the 
least that can lie given to a pinion. Now, in any given case, 
let us suppose L,„ greater than L^, so that the wheels will be 
the drivers, and the pinions the followers. The least number 
of axes will then evidently be obtained by giving each wheel 
10 teeth, and each pinion 79 teeth. The number of axes being 
m, we will have (Art. 222) m — 1 wheels and m — 1 pin- 
ions. Hence 



^1 X ^2 X -^3 X • • • ^/^ 



A 






^2 


X 


n^ X 


n^ X 




• ^m 




w 


X 


w 


X 


IV to 


{711 — 


1) 


factors 




P 


X 


p 


X 


p to 


(m — 


1) 


factors 


whence log e 


__ 


(m - 


- 3) 


(logw 


_ 


logp), 






loSf 6 

.-. m = 1 + 1 ^-T (6) 

log IV — logp ^ ' 

The least number of axes, under the assigned conditions of 
%v and p, is evidently the value of m thus found, if this value 
is a whole number ; or the whole number next larger than 
this value of m, if the latter is fractional. No general rule 
can be given for determining the values of %v and p^ which 
are governed by considerations that vary according to the 
nature of the proposed machine ; also, it will rarely happen 



TRAINS OF MECHANISM. 251 

that the fraction will admit of being divided into factors so 
nearly equal as to limit the number of axes to the smallest 
value so assigned, 

232. Practical Example of Clock Train. — We will 
now return to the consideration of the clock described in 
Art. 226, and show how the number of axes and the number 
of teeth of the wheels and pinions were determined. It was 
required that the first or barrel axis should revolve once per 
hour, and that the m^^^ or swing- wheel axis should carry a 
seconds hand, S. The swing- wheel axis must therefore 
revolve once per minute, or sixty times per hour. 

Consequently 



A. 


60 

" T ~ 


iVj X -^2 X . . . jsr„,_i 


"A ■ 


11^ X 71^ X , . . n^ 



Let D be the numerator of this fraction, i.e., the continued 
product of all the drivers, and let F be the denominator, i.e., 
the continued product of all the followers. 
Then 

c = 60 = ^ .-. i> = 60 X i^, 
F 

an indeterminate equation, for the solution of which any 
numbers may be employed that are within the assigned limits 
of IV and p. Now, in ordinary clocks, iv = 60, and ^9 = 6, 
so that 

H = 60 ^ ^^^ 

p 6 

From Equation (5) , we have 

e = 60 = (10)"*-!. 

We can then determine the value of m by means of Equation 
(6) ; or, what is much simpler, determine, by inspection, the 



252 ELEMENTARY MECHANISM. 

value of m — 1 from the above expression. The latter 
method is to be preferred, as tlie exact value of m — 1, if it 
be fractional, is of no consequence, it being simply necessary 
to determine the next greater whole number. 

Thus, in our example, it is evident, as 60 lies between 10^ 
and 10^, that m — 1 must lie between 1 and 2, consequently 
m must lie between 2 and 3 ; and, taking the next larger 
whole number, we fix on m = 3, as the least number of 
axes. Consequently there will be two wheels and two pin- 
ions. Taking the pinions at six teeth each, we have 



e = 60 = ^ = ^ 



F 6x6 

.-. D = 60 X 6 X 6 = 2160, 

which is the product of the two wheels. 

We are at liberty to divide this into any two suitable fac- 
tors. The best mode of doing it is to begin l)y dividing the 
number into its prime factors, and writing it in this form, 

2160 = 2x2x2x2x3x3x3x5. 

For this enables us to see clearly the composition of the 
number, and it is easy to distribute these factors into two 
groups ; as, for example, 

(2x2x2x2x3) X (3 X 3x5) = 48 X 45, 

or 

(2x2x2x5) X (2 X 3 X 3 X 3) = 40 X 54, 

or 

(2x2x3x8) X (2 X 2 X 3 X 5) = 36 X 60. 

The first group will give us the two wheels that are most 



TRAIL'S OF MECHANISM. 253 

nearly equal, whick is a sufficient reason for selecting that 
pair for our train. We now have 

D ^ 48 X 45 
F Q X Q ' 

So far we have only determined on the numbers of the 
teeth of the various wheels, without locating them as regards 
the different axes ; and the above fractional expression is an 
excellent method of exhibiting the train under these condi- 
tions. 

As before stated, the order in which the wheels come is a 
matter of indifference, so far as the velocity ratio is con- 
cerned ; and, as no other considerations enter into this case, 
we will place driver 48 on the first axis, follower 6 and driver 
45 on the second axis, follower 6 and swing-wheel 30 on the 
third axis, giving us, as in Art. 229, the train 

48 
6 — 45 

6 — 30. 

233. Another Clock Train. — Six is, however, too 
small a number of leaves for the pinion, if perfect action is 
desired ; for it is evident, from the table of Art. 134, that a 
pinion of 6 teeth cannot drive a wheel of less than 21 teeth, 
if the arc of recess equal two- thirds pitch ; while, if this arc 
is increased to three-fourths pitch, a pinion of 6 cannot be 
made to work at all. In well-made clocks, p is generally 
taken between 8 and 12, while w ranges from 100 to 120. 

Let us find a new train for our clock, having p = 12, and 
tv = 105. 

We have 

c= 60 = (i%5).«-i=(8.75)' 



rr;\m— 1 



from which we see, by inspection, that the value of m — 1 is 



254 ELEMENTARY MECHANISM. 

fractional, and lies between 1 and 2 ; that the value of m lies 
between 2 and 3 ; and that the least number of axes will 
consequently be 3. Assuming the two pinions to be equal, 
and to have the smallest allowable number of teeth, we have 

D N y. N 

F = 12 X 12 = ^^ •*• ^ = 60 X 12 X 12 = 8640. 

Proceeding as in the last example, we find the best values 
for the wheels tobei)= 96 x 90. We then have 

D ^ 96 X 90 
i^ 12 X 12' 

and, placing them on their axes, we have the train 

96 

12 — 90 
12. 

Instead of assuming the pinions, we might have started with 
the wheels. Thus let us take 

D ^ 105 X 105 ^ gQ 

F n.^ X Wg 

... F = 12LX^2£ ^ 183.75. 
60 

It is evidently impossible to divide 183.75 into two integer 
factors ; and, as we cannot increase the assumed number of 
teeth for the wheels, we must diminish the number of one or 
both. Let us take one of the wheels as 104. This will give 

us 

104X105^ 

GO . - 



TRAINS OF MECHANISM. 255 

which can readily be factored, giving us F = 13 x 14, and 
the train 

104 
13 — 105 
14. 

It very often happens, as just illustrated, that attempting to 
make the wheels and pinions with the limiting numbers of 
teeth gives rise to very awkward results, while an excellent 
train can, in such cases, be generally found by trying several 
numbers within the limits. 

234. Clock with rapidly vibrating- Peiululum. — 
If a clock has no seconds hand, the limitation as to the 
period of one revolution of the swing-wheel axis is removed. 
This is an advantage in clocks having short, and conse- 
quently rapidly vibrating, pendulums ; for it would be imprac- 
ticable to make the period of the swing-wheel axis one 
minute, as before, on account of the great number of teeth 
which would be required for the swing-wheel. If ^ = time 
of vibration of the pendulum in seconds, and A == number 
of teeth in the swing-wheel, then (as in Art. 226) 2^A is the 
time required for one revolution of the swing- wheel. 

But the vibrations of short pendulums are commonly ex- 
pressed by stating the number of them in a minute. Let S 

2A 
be this number ; then -— ■ is the time of one revolution of 

S 

the swinaf-wheel in minutes ; — is the number of revohi- 
° 2A 

tions of the swing-wheel axis per minute ; and, as the barrel 

arbor revolves once per hour, we have for the train between 

them, 

^ ^ D ^ eOS ^ SOS 

For example, let the pendulum of a clock make 170 vibra- 



256 ELEMENTARY MECPIANISM. 

tions per minute ; let there be 25 teeth on the swing-wheel ; 
then 

, = :? = 30 X 170 ^ 
F 25 



Taking w = 128, and p = 8, we have 
w 128 -^ 

and, as 204 = (IGj'^-S we see, by inspection, that the least 
number of axes is 3. 

Assuming the pinions as each having 8 teeth, we have 

i> = 204 X i^ = 204 X 8 X 8 = 13056 = 128 X 102. 

Hence the train is 

128 
8 — 102 

8 — 25. 

235. Eigrht-Day Clock. — All the trains so far ex- 
plained were designed to establish the proper velocity ratio 
between the hour arbor and the swing-wheel axis. It was 
assumed in each case that the hour arbor also carried the 
weight-barrel ; and, as we limited the number of coils of the 
cord to 16, it follows that the clocks so far considered will 
only run 16 hours without re- winding. 

If we adhere to the limitation as to the number of coils 
of the cord, but still desire the clock to run longer than 16 
hours, the barrel must be attached to a separate axis con- 
nected by wheel-work with the hour arbor, so that the barrel 
may revolve more slowly, consequently taking more time to 
uncoil all the cord. 

For example, let the clock be required to go 8 days with- 



TRAINS OF MECHANISM. 257 

out re- winding ; then, with 16 coils of cord on the barrel, the 

latter must revolve once in • = 12 hours. Then, as- 

16 

suming w = 100, and p = 8, we may use the train, — 

Periods. 

Barrel arbor, 96 12 hours. 

Hour arbor .8 — 90 1 hour. 

12 — 96 8 minutes. 

Minute arbor . . . .12 — 30, swing- wheel . . 1 minute. 



It is often convenient to add to the notation the periods of 
the different arbors, as has been done in this case. 

236. Month Clock. — Let the clock be required to run 
32 days without re- winding, and let there be 16 coils on the 
barrel as before ; then the latter must revolve once in 

= 48 hours. The train from the barrel to the 

16 

hour arbor is — = 48, which will require an intermediate 

axis. 

Letting iv = 100, and p = 12, we may employ the follow- 
ing train : — 

Periods. 

Barrel arbor, 96 48 hours. 

16 — 96 8 1iours. 

Hour arbor . . 12 — 90 1 hour. 

t 12 — 96 8 minutes. 

Minute arbor 12 — 30, swiiiff-wheel, 1 minute. 



237. Now, in the clock (Fig. 169), the arbor of A is 
made to revolve in one hour, because the wheels E and e are 
equal. By making these wheels of different numbers, we 
get rid of the necessity of providing, in the principal train, 
an arbor that shall revolve in one hour ; and we may thus, 
in many cases, distribute the wheels more equally. For ex- 
ample, in an eight-day clock let the swing-wheel revolve once 



258 



ELEMENTARY MECHANISM. 



per minute, and let the train from the barrel- arbor to this 
minute-arbor be 



D ^ 108 X 108 X 100 
F" 12 X 12 X 10 



= 810, 



in which case the barrel will revolve once in 810 minutes, or 
13 J hours. 

The second wheel of this train, which, in Fig. 169, cor- 
responds to D^ will revolve in -f-^^ x 810 = 90 minutes, or 
1^ hours. On its arbor must be fixed, as in the figure, the 
wheels E and F for the minute and hour hands ; and we 
may employ, for the two pairs of wheels. 



F 

1 



12" 



1 



10 

80 



and 



F 



1 



54 
3G' 



So that our train will be as follows : — 

Periods. 

Barrel, 108 810 minutes. 

12 - 108 54 10 ... 90 '« 



12 -- 100 



10-30 pwJ^g- . 
( wheel ' 



minute- 
hand 



hour- 
hand 



10 «' 
1 minute. 

1 hour. 

12 hours. 



238. The above examples have been confined to clock- 
work, because the action is more generally understood than 
that of other machines. The principles and methods are, 
however, universally applicable, or, at least, require very 
slight modifications to adapt them to particular cases. 

For instance, in a screw-cutting lathe, there is usually one 
intermediate axis between the leading-screw and the head- 
stock spindle. Let the leading-screw be right-handed, and 



TRAINS OF MECHANISM. 259 

have two threads to the mch ; let iv = 130, x> = 20 ; and 
let it be required to cut a right-handed screw of 13 threads 
to the inch. Here 

^ ^ ^ 13 ^ 130 X 90 
^ ~ i^ ~ 2 ~ 20 X 'JO ' 

which is a good train for the purpose. The wheels for form- 
ing a series of such trains, calculated for the different numbers 
of threads to be produced, are known as a set of cJiange- 
wlieels ; and tables for the use of such wheels are furnished 
by lathe-manufacturers with all screw-cutting lathes. 

239. Frequency of Contact between Teeth, — It is 
sometimes a matter of interest to know how often any two 
given teeth will come into contact as the wheels run upon 
each other. We will take the case of a wheel of A teeth 
driving one of B teeth, where A is greater than 5, and let 

A a 

— = - when reduced to its lowest terms. 

B b 

It is evident that the same points of the two pitch circles 
would be in contact after a revolutions of B, or b revolutions 
of A. Hence, the smaller the numbers which express the 
velocity ratio of the two axes, the more frequently will the 
contact of the same teeth occur. 

1. Let it be required to bring the same teeth into contact 
as ofteyi as possible. 

Since this contact occurs after b revolutions of A, or a 
revolutions of B^ we shall effect our object by making a and 
b as small as possible ; this is, by providing that A and B 
shall have a large common divisor. 

For example, assume that the comparative angular velocity 
of the two axes is intended to be as nearly as possible as 5 
to 2. Now make A = 80, B = S2; then 



^ 80 5 .. 

5 = 32 = 2'"^'^'^' 



260 ELEMENTARY MECHANISM 

or, the same pair of teeth will come in contact after 5 revo- 
lutions of jB, or 2 of A. 

2. Let it be required to bring the same teeth into contact 
as seldom as possible. 

Now chansje A to 81, and we shall have — = — = - vei'v 

B 32 2 

nearl}^ ; or, the angular velocity of A relativel}^ to B will be 

scarcely distinguishable from what it was originally. But 

ft 81 
the alteration will effect what we require, for now - = — . 

^ & 32 

There will, therefore, be a contact of the same pair of teeth 

only after 81 revolutions of J5, or 32 revolutions of A. 

The insertion of a tooth in this manner was an old contriv- 
ance of millwrights to prevent the same pair of teeth from 
meeting too often, and was supposed to insure greater regu- 
larity in the wear of the wheels. The tooth inserted was 
called a hunting cog, because a pair of teeth, after being 
once in contact, would gradually separate, and then approach 
each other by one tooth in each revolution, and thus appear 
to hunt each other as they went round. 

Clockmakers, on the contrary, appear to have adopted the 
opposite principle ; though it has probably been partly forced 
on them, as the velocity ratio of the clock arbors must neces- 
sarily be exact. 

240. Approximate IS^umbers for Trains. — If -^ 

= kj when A; is a prime number, or one whose prime fac- 
tors are too large to be conveniently employed in wheel- 
work, an approximation may be resorted to. For example, 

assume -^ = k ± h. This will introduce an error of ±h 

revolutions of the last axis during one of the first, and the 
nature of the machinery in question can alone determine 
whether such a variation is permissible. 

For example, let e = -^ = 269, which is a prime num- 



TRAINS OF MECHANISM. 261 

ber. Take € = 269 -f 1 = 270, which can readily be fac- 
tored into 6x5x9; and we may employ the train 

i> 72 X 60 X 90 rr., . > • •,, . 

— = . Ihis tram will cause an error of one 

F 12 X 12 X 10 

revolution of the last axis for every revolution of the first 

axis, the altered value of e varying less than two-fifths of one 

per cent from the correct value. 

241. But we may obtain a better approximation than this, 

without unnecessarily increasing the number of axes in the 

train ; for, determine, in the manner already explained, the 

least number m of axes that would be necessary if k were 

decomposable, and the number of teeth that the nature of 

the machine makes it practicable to give to the pinions, and 

let F be the product of the pinions so determined ; hence 

L^ F F 

supposing the wheels to drive. 
Assume 

D ^ Fk ± h 
F F ' 

where h must be taken as small as possible, but so as 
to obtain for Fk ± h si numerical value decomposable into 
factors. There will be, in this case, an error of ±h revolu- 
tions in the last axis during F of the first, or an error of 

— during one of the first. If the pinions are to be the 
drivers, then, in the same manner, assume 

ii _ Dk ± h . 

l:" d ' 



and there will then be an error of -— ^ revolutions in the first 



262 ELEMENTARY MECHANISM. 

axis during one revolution of the last axis. Let us take, as 
in the previous example, e = 269. Let w = 90, and p = 
10; then 

269 = (9)"*-^ 

whence we find the least number of axes to be four. 

Let us assume that pinions of 10 will be employed ; then 



€ = - = 269 = 269000 



F 10 X 10 X 10 

Now add 1 to the numerator, and we have 



D 269001 


81 X 81 X 41 


F 10 X 10 X 10 


10 X 10 X 10 



This will give a good train with an error of only 1 revolution 
in 269000. 

As another example, let it be required to find a train that 
shall connect the twelve-hour wheel of a clock with a wheel 
revolving in a lunation (viz., 29 days, 12 hours, 44 minutes 
nearly) , for the purpose of showing the moon's age on a dial. 

Reducing the periods to minutes, we have 

Lm 42524 
, L^ 720 ' 

of which the numerator contains a large prime ; viz., 10631 ; 
but 

42524 + 1 ^ 60 X 63 
720 8x8' 

giving a good train, with an error of one minute in a lunation. 



AGGREGATE COMBINATIONS. 263 



CHAPTER XIII. 

AGGREGATE COMBINATIONS. 

Differential Pulley. — Differential Screw Feed Motions. — Epicyclic 
Trains. — Parallel Motions. — Trammel. — Oval Chuck. 

242. Ag-g-regate Combinations is the term applied to 
those assemblages of pieces in mechanism in which the 
motion of a follower is the resultant of the motions it re- 
ceives from more than one driver. The number of drivers 
which impress their motion directly upon the follower is 
generally two, and cannot exceed three, since each driver 
determines the motion of at least one point of the follower, 
and the motion of three points in a body determines its 
motion. 

Such combinations enable us to produce by simple means 
very rapid or very slow velocities^ and complex paths, which 
could not well be obtained directly from a single driver. 
These combinations may be divided into two classes, accord- 
ing as velocity or path is the principal object to be attained ; 
7iind we will consider these two classes separately. 
i 

Aggregate Velocities. 

243. By Linkwork. — In Figs. 170 and 171, let AB be 

a rigid link, and let the point A be given a velocity a, while 
the point B is given the velocity b. Then it is required to 
determine the motion of an intermediate point, C, which is 
affected by the motions of both A and B. These motions 
are generally perpendicular to AB, or so nearly so that the 



^64 



ELEMENTARY MECHANISM. 



error in their comparative motions will not generally be prac- 
tically appreciable. 

jB' 

1b 




B^ig.lT'O 



If we consider the motion of A alone, regarding B as 

BC 

stationary, C will move with a velocity = ^^•~;r~* ^^ ^^ ^^^^' 

sider the effect of the motion of B alone, regarding A as 

AO 



stationary, we have the velocity of C = b. 



AB 



Considerina: 



motion in one direction as positive, and in the opposite direc- 
tion as negative, we have for the resultant motion of C from 



both A and B, c 



a.BC 4- b.AC 



AB 



, or the algebraic sum of the 



two component velocities. 



A 


C B 






I 


a 


G 




A' 







jrig.171 



This result may be represented graphically, as follows : 
Perpendicularly to AB draw AA and BB' to represent in 
length and direction the velocities of A and B respectively. 
Draw AB\ Then CC drawn through C perpendicularly to 
AB will represent in length and direction the resultant velo- 
city of the point C. 

Examples of aggregate motion by linkwork are to be seen 



AGGREGATE COMBINATIONS 



265 



ID the several forms of " link motion " valve gears of revers- 
ible steam-engines. In these, motion is given by eccentrics 
or cranks to points such as A and B in the figures, and the 
steam-valve receives its motion from some intermediate point, 
the distance of which from the ends can be varied. As will 
be seen from the figures, if C is nearer A than B, for instance, 
its motion will be derived to a greater extent from A than 
from B. If it is midwa}^ l)etween these points, it will re- 
ceive an equal proportion from each. 

244. Differential Pulley. — In Weston's differential 
pulley, illustrated by Fig. 172, the principle of aggregate 




^66 ELEMENTARY MECHANISM. 

velocities is made use of for lifting heavy weights by the ap- 
plication of a small amount of force. It consists of a single 
movable pulley, D, from the axis of which the weight to be 
lifted is suspended ; a fixed pulley, C, having two circum- 
ferential grooves, the diameter of one being somewhat less 
than that of the other ; and an endless chain passing around 
the pulleys, as shown in the figure. The combination is ope- 
rated by hauling upon the chain LN in the direction indicated 
by the arrow. The velocity of the pitch circle, EL, is evi- 
dently equal to that of the hauling part of the chain. Let 
I, k, denote the velocities of the pitch circles EL and HK 
respectively, and h the velocity of BP. 

Then, if the point K were stationary, hauling down upon 

LN would evidently raise B with a velocity = -. But K^ 

being rigidly connected to X, moves downward with a velocity 

k A^K A.K 

such that - = — — , or k = I.— — . Considering E as fixed, 
I x\.L x±L 

k 
this would give to B a downward velocity of -. Hence the 

resultant velocity of B upwards will be 

fe =^ — ^= 7 ^L - AK 
2 2 * 2AL ' 



or the velocity ratio = - = ^^ ~ ^^^ . 
^ I 2AL 

245. Compound Screws. — In Fig. 173 let SS' be a 

cylinder upon which two screw threads are formed. Let the 
portion ah have a pitch n, and be fitted in a fixed nut iV; 
and let the portion cd have a pitch m, and be fitted with 
a nut Jf which is free to move in the direction SS', but which 
is prevented from turning. Then, if the bolt be turned in 
the nuts as indicated, it will move through the nut JSf, a dis- 
tance n, during each turn, while at the same time the nut M 



AGGREGATE COMBINATIONS. 



267 



will move along ^S'-^S", a distance ??i, during each turn. There- 
fore, if the screws wind the same way, M will move relatively 
to the fixed nut jV, a distance equal to the difference between 
n and m for each turn of SS' . That is, if n is greater than 
wi, M will move awajj from N the distance ii — m for each 




Fig. 173 

turn ; or if m is greater than 7i, M will move toivards JSf 
the distance m — n. If the screws wind in opposite ways, the 
motion of M relatively to JSf will be n + m for each turn. 

246. Automatic Drill Feed. — Fig. 174 illustrates a 
combination for the production of a slow endlong motion 
of a spindle, together with a rapid rotation such as is re- 
quired for the spindle of a drill-press. In the figure, AB is 





K 


K 









ci 






ID 









1 






1 




N 




Ai M \\\\\^ 


\\\\\ iB 




Ie 


F 





Fig. ±74= 



the spindle to which is fastened the spur wheel E. A thread 
is cut on a portion of AB, to which is fitted a nut N mounted 
in the frame of the machine, so that it is free to rotate, but 
can have no other motion. To A" is fixed a spur wheel F. 
E and F gear respectively with a long pinion H and a spur 
wheel /r, both fixed to a driving-shaft CD. Let c be the 



268 



ELEMENTARY MECHANISM. 



number of revolutions made by CD, while F and E make 
/ and e revolutions respectively. Also, let E^ F, H, and K 
represent the number of teeth upon the respective wheels. 

Then, - = — , and -^ = — . Let^ be the pitch of the screw, 
then c revolutions of CD will cause AB to travel through the 



distance (/ — e)j^j = cpl j. 

\F E J 



TT J7- 

For example, let p = J'', — = y^q , and — = J ; then, for 

E F 

one turn of CD, the spindle will travel J''(f - ^-^)=l"x ^V 

— -i-J^ 

— 80 • 

247. An Epicyclic Train is a train of mechanism, the 
axes of which are carried by a revolving arm. Simple 



Si ii J i lii i iii il 



7" 



forms of epicyclic trains are illustrated by Figs. 175 and 
176. In both figures the train-bearing arm. A, revolves 
about a fixed centre, jB, and carries the train of wheels 
shown. (7, which is considered to be the first wheel of the 



iiiiiiiiiittiiiiii ■iiii.it^iiiiiiiiiii 



iiiiiii!c|iiiiiiiiiiiii [jfiiiii 



Fis. 17'6 



train, is concentric with JB, and may be fixed, or may receive 
motion from some external source. The wheel E, which is 
considered to be the last wheel of the train, may be carried 
by the arm, as in Fig. 175, or be concentric with it, as in 
Fig. 176. In the latter case it is carried by a separate shaft, 



AGGREGATE COMBINATIONS. 



269 



or turns loosely upon B. In either case its actual motion is 
the resultant of the motion derived from the revolution of the 
arm A and that received from C by means of the connecting- 
train. It will be seen that the connection between C and E 
]nay be made by any of the modes of transmitting motion 
w^hich have been discussed. 

Epicj^clic trains are used: (1) To produce an aggregate 
motion of the last wheel by means of simultaneous motions 
given to the first wheel and the arm. (2) To produce an 
aggregate motion of the arm by means of simultaneous mo- 
tions given to the first and last wheels. 

248. Velocity Ratio in Kpi cyclic Trains. — In Fig. 
177 let A be the train-bearing arm of an epicyclic train 
turning about B. Let C be the wheel concentric with 5, 




Fig. 177 



and E the axis of a wheel F carried by the arm and con- 
nected to (7 by a train of mechanism. Suppose that while A 
turns about B to some other position A\ a point a, on wheel 



270 IlLEMENTARY MECHANISM. 

C, moves to h from any external cause, and that a point d, 
on wheel F^ moves to e by reason of its connection with C. 
For simplicity, all are supposed to turn in the same direction. 
Draw T^'F parallel to EB. Then aBh and liE'e are the 
absolute angular motions of C and F respectively, and cBh 
and gE'e are their angular motions relatively to the arm A. 

JiE'g = aBc — angular motion of the arm. 
aBh = aBe + cBb. 
liE'e = liE'g + gE'e = aBc + gE'e. 
Or, 

cBb = aBh - aBc; gE'e = liE'e - aBc. 

These equations are true for angles of any magnitude, and 
hence for complete revolutions since the velocity ratio is con- 
stant. 

Let a, m, and n be the synchronal absolute rotations of the 
arm, of the first wheel O, and of the last wheel -F" respectively. 
Let € be the value of the train between G and F^ that is the 

quotient which has been represented by — ^^ = — in Chap. 

Ly F 

XII. Then the rotations of the first wheel relatively to the 
arm = m — a, and the rotations of the last wheel relatively 

to the arm = n — a. Therefore € = , which is the 

m — a 

general equation for epicyclic trains. 

From this we derive 



a = — , m = a -] , n = a + €(m — a). 

C — 1 € 

If the first wheel is fixed, m = 0. 



a — n n /i \ 

€ = , a = , n ={1 — €)a. 



a 1 — € 



AGGREGATE COMBINATIONS. 



271 



If the last wheel is fixed, n — 0. 
a me 



a — m 



a 



1 



=(-iy. 



In all of the above formulae, the arm, 



last 



wheel are assumed to rotate in the same direction ; but if the 
direction of rotation of any one is changed, the sign of a, 
m, or n should be changed accordingly. In applying the 
formulae, we first assume that the rotations take place in the 
same direction, and then, one direction for the arm being 
taken as positive, the -h or — sign of m and n will show 
whether they are rotating in the same direction or the reverse. 

If the connecting train is such that the first and last wheels 
would rotate in the same direction, supposing the arm to be 
fixed, the sign of e is phis, but if they would rotate in 
opposite directions, it is to be taken as minus. For example, 
if the connection is by spur gearing, and there are an odd 
number of axes, e is positive; but if the number of axes is 
even, e is negative. 

249. Ferguson's Paradox, illustrated by Fig. 178, will 
serve as a shnple example for the application of these formu- 

H 

IIIIIIIIIILIIIIIIIIilll lTrrrrT 

Id, 




inig.i7S 



Ise. The wheel C has 20 teeth, and is fixed to the shaft B, 
about which the arm A rotates. This arm carries the axis 
of the "wheel D, which gears with C and with three wheels 
E, F, and 6r, which turn loosely on the shaft H also carried 
by the arm. E has 19 teeth, F 20, G 21, and D any num- 
ber. Since there are three axes, e is -|-, and has the three 
20 -, O 20 



, O 20 O 

values, — = — , — 

'E I'd F 



, and — = 
20 G 21 



C is fixed ; there- 



fore, m = 0, and n ={\ — €)a. 



272 ELEMENTARY MECHANISM. 

In the three cases we have 

(ff) „ = (l - |)„ = +1 „. 

That is, when the arm revolves the wheel F will have no 
absolute rotation, while, for each revolution of the arm, E 
will make -^^ of a turn in the opposite direction, and G will 
make ^^y of a turn in the same direction. 

250. Watt's Crank Substitute, otherwise known as the 
jSiin and Planet Motion^ belongs to the general class of epi- 
cyclic trains. In Fig. 179, AB is one end of the main beam 
of an engine, (7 is a spur wheel fastened to the main shaft, 
and ^ is a spur wheel fastened to the connecting-rod BD^ 
and gearing with C. E is held in gear with G by means of 
a connecting link OD, or by a circular groove concentric 
with C in which a pin at D slides. As E is raised and 
lowered by the motion of the beam, and forced to revolve 
about (7, since it cannot rotate its own axis, it causes C 
to rotate. E has a vibratory motion due to the varying 
angle of the connecting-rod, but as this is periodic, it may 
be neglected for complete revolutions. 

Considering the combination as an epicyclic train, OD will 
be the train-bearing arm, C the first wheel, and E the last 
wheel. The latter has no absolute rotation ; hence, applying 

the general formula, and letting n = 0, we have m = a( 1 J. 

Mso, since there are but two axes, 



Let G = E, then € = -1, m = a(l i-^ = 



2a, 



AGGREGATE COMBINATIONS. 



273 



Or, for one revolution of the train arm OD corresponding to 
an up-and-down stroke of the piston, C makes two revolu- 
tions. Thus by this arrangement the shaft rotates twice as 
fast as it would with the ordinary crank connection. If C has 

twice as many teeth as E^ e = —2, and m 



i' - -^) 



= -«, or C revolves three times while OD revolves twice. 



If E has twice as many teeth as O, e = 



m = a(l -f 2) 



= 3a, or C revolves three times for one revolution of OD. 




251. Epicyclic trains are used in some forms of rope- 
making machinery. In order that a rope shall not untwist, 
it is necessary that the separate strands shall either be laid 
together without any twist, as in wire rope, or that they shall 



274 ELEMENTARY MECHANISM. 

have a slight twist in the opposite direction to tlie apparent 
twist of the rope. In Fig. 180, let B be the bobbins from 
which the wire or strands are unwound as the rope is formed. 
These bobbins are carried by wheels D, which are connected 
to a centre wheel A by intermediate wheels C. The axes 
of all the wheels excepting A are carried by a frame which 
turns about the axis of ^. If the bobbins were fixed in 




this frame, as the frame revolved, each strand would be 

twisted as it was unwound, but if we arrange it so that the 

axes of the bobbhis shall always lie in the same direction, 

there will be no twist. This is accomplished by fixing the 

axes of the bobbins to the wheels i), fixing the wheel A^ and 

making D = A. We then have an epicyclic train in which 

r. T n — a AC ^ 

m = 0, and e = = — x — = 1, .' . n — a = —a, 

— a CD 

and ?i = 0, or the wheels D have no absolute rotation, and 

consequently there is no twist given to the strands. By giving 

D a few more teeth than A^ the strands will be given a slight 

twist in the opposite direction to the twist of the rope. 



AGGREGATE COMBINATIONS. 275 

252. Epicyclic trains may be used to transmit velocity 
ratios which could not be conveyed by direct trains except 
by using a large number of axes or inconveniently large 
wheels. The necessity for such ratios rarely arises except 
in astronomical machinery, and for explanations of such 
applications the student is referred to Willis' "Principles 
of Mechanism," and the works there referred to. 

Aggregate Paths. 

253. Parallel Motions. — The most important applica- 
tion of aggregate combinations in which the iKith is the 
immediate object sought, is to give motion to a piece such 
that a point in it shall move in a straight line. Such combi- 
nations are commonly called "parallel motions," although 
' ' straight-line motion ' ' would be a more correct and de- 
scriptive name. 

Some of these combinations give an exact straight-line mo- 
tion, but in most of them the motion is only approximate. 
We have seen an example of exact straight-line motion 
in the case of a point on the circumference of a circle roll- 
ing within another circle of twice its diameter, being in fact 
a special case of the hypocycloid. By means of accurately 
cut gears, this could, of course, be applied to machinery. 

In the parallel motions in general use, the straight-line path 
is produced by combinations of links, and such combinations 
will be now considerecl. 

254. Peaucellier's Exact Straiglit-Line Motion. — 
In Fig. 181 is shown the general arrangement of Peaucel- 
lier's exact straight-line motion. It consists of seven mov- 
able links connected as shown. Two long links AD^ AE, 
oscillate about a fixed centre yl, and are jointed at the ends 
D and E to opposite angles of a rhombus, CDPE, composed 
of four shorter links. At C is connected a link J5C, oscil- 
lating about a fixed centre B^ so located that AB = BC. 



276 



ELEMENTARY MECHANISM. 



Then the point P will describe a straight line perpendicular 
to AB. 

D 




B iPig.isi 



From the symmetrical construction of the combination it 
is evident that the points A^ (7, and P must always lie in 
one straight line. Let the combination be moved, Fig. 182, 




from the central position shown dotted, to some other posi- 
tion, such as that shown in full lines, the point F occupying 
the position P'. Draw AP, AP^, and CC ; also DL per- 



AGGREGATE COMBINATIONS. 277 

pendicular to AP^ and B'K perpendicular to AP' , From the 
construction, P'K = KC , and PL = LC. Then, 



AD'' = Air + KD'" = ^^' +(Z)'C" - KC) ; 



=^{AK- KG'){AK^ KC) = ^0' x AP\ 
Similarly, 

zd' = zl' + :dz' = 3Z' + (:dc' - zo') ; 

^{AL - LC){AL + iC) = AG X AP. 

. • . ^C X ZP = ^C" X ^P' ; 
or 

AP ^ AC 
AP' AC ' 

AC is a diameter of the circle ACC ; hence CCA is a right 
angle, and P'P is perpendicular to ABP. And P' having 
been assumed as any position of P, it follows that the above 
relation is true for all positions, or P moves in a straight line 
perpendicular to AB. 

255. In applying this motion to engines, the point P is 
connected to the end of the piston-rod, and thus takes the 
place of the usual cross-head and guides. It is to be par- 
ticularly noted, that, as stated above, the arm BC is equal in 
length to the distance AB. If this is not so, instead of a 
straight line, circular arcs will be described by P. If the 

ratio — — is less than one, the arc described will be concave 

towards A ; if the ratio is greater' than one, the arc described 
will be convex towards A ; and if the ratio is eqiial to 07ie, 
the circular arc becomes a straiqlit line. 



278 



ELEMENTARY MECHANISM. 



There are other exact parallel motions * formed by combi- 
nations of linkwork, most of which are derived from the 
Peaucellier cell ; but they are of so little practical impor- 
tance that they will not be discussed in these pages. 

256. Watt's Approximate Straight-Line Motion. — 
The most widely used of the approximate straight-line mo- 
tions is that invented by James Watt. It is shown in its 
simplest form in Fig. 183. AC and BD are two arms 




Fig. 183 



turning about fixed centres A and B^ and connected by a 
link CD. When in the mid position the arms are parallel, 
and CD is perpendicular to them. If the arms be made to 
oscillate, a point in 0J9, such as P, will describe a figure 
similar to that shown. But we can so arrange the propor- 
tions of the links, and the position of P, that for a limited 
motion it will not deviate much from a straight line. 

257. Let the arms AC and BD be turned to some other 
positions, as Ac and Bd in Fig. 184. Then the link CD will 
be moved to cd. The end C has been moved to the right, 
and the end D to the left, so there will be some point P, of 
cd^ which will lie in the continuation of the line CD. Let 



* For description of parallel motions referred to, see A. B. Kempe's 
"How to Draw a Straight Line." See also American Macliiuiat, Sept. 
17tli, 24th, Oct. 1st, 15th, 33d, 39th, and Dec. 3d, 1891. 



AGGREGATE COMBINATIONS. 



279 



AC =R,BD = 7-, CAc = 0, DBh = </>, CD = Z, and cP = x. 
Drawing ce and dg parallel to AC^ we have 



cP _ X _ ce_ _ B(l — cos 0) 
dP I — X dg r (1 — cos ^) 



2i2sin2 



i^^sin^: 



2r sin^ 



</> Pi, 



r^ sin^ * 



In practice, ^ does not exceed about 20°, the inclination of 
the link cd is small, and RO is very nearly equal to r<^. As 

these angles are small, we may assume i? sin - = r sin — , 




Fig.lS^r 



hence = — , or the seorments of the link are inversely 

I -X R ^ ^ 

proportional to the lengths of the nearest arms, which is the 
usual practical rule. 

258. Amount of Deviation. — The deviation of the 
point P from the line /S'/S' can be calculated, but will not 
generally exceed about -^ inch. This may be greatly re- 
duced by the arrangement shown by Fig. 185, which should 
always be used. In the mid position the arms are perpen- 
dicular to the line &S in which the point P should lie, and 



280 



ELEMENTARY MECHANISM. 



which in an engine should coincide with the centre line of 
the cylinder or pump. This line should bisect the distances 
Ce and Df which are the versed sines of the maximum values 
of the angles and <^. The ends C and D of the link will 
then evidently deviate equal amounts on each side of SS. 
Drawing dli and dg perpendicular to SS^ and Cn parallel 
to SS^ we have three equal triangles, cWi, CDn^ and cdg. 
Therefore, c'p' = GP = cp, or the mid and extreme posi- 
tions of the guided point P are exactly on S8, 




The greatest deviation of the guided point from S8 occurs 
when CD is parallel to SS, and is best determined in any 
case by drawing the combination to a large scale, and find- 
ing the parallel position by trial. 



AGGREGATE COMBINATIONS. 



281 



259. Problem. — In Fig. 18G, let CA be an arm as 
before, cA its extreme position, and SS the line of stroke bi- 
secting Ce. Join (7c, and draw AN perpendicular to it. iV 
bisects Cc, since the latter is the chord of the angle CAc, 
and hence is on the line SjS. Also MJSf = i ec, or, since ec 
may be taken as | the stroke, MJSf = J the stroke. 




Therefore, if we have given the length of stroke and direc- 
tion, SS, the centre of one arm A^ and mid position of the 
guided point P, we can construct the remainder of the mo- 
tion as follows : Draw AR perpendicular to SS, lay off MJSf 
= J stroke, draw AN, and perpendicular to the latter draw 
NC. Where this line intersects AR at (7, will be the end of 
the arm AC. CP will be the direction of the link in mid 
position. If we assume, or have located, the point H where 
the mid position of the second arm cuts SS, draw an indefi- 



282 ELEMENTARY MECHANISM. 

nite straight line, FH^ through this point perpendicular to 
SS. The point D, where CP produced cuts FH^ is the ex- 
tremity of the second arm. Then, since HD must be ^ the 
versed sine of the arc through which D moves in either direc- 
tion, we can find the centre B by laying off HT = J stroke, 
and drawing TB perpendicular to TD. 

260. Practical Form of Watt's Motion We have 

thus found the proper proportions for the simplest form of 
the motion ; but, as usually constructed, the motion is of the 
form shown in Fig. 187. AE is one arm of the main beam 




B^ig.187 



of an engine, and turns about the centre A. EF is the 
main Ihik^ connecting AE with the piston-rod F8. CD is 
the hack-link equal and parallel to EF. FD is the parallel- 
bar equal and parallel to EG. BD is the radius bar, or 
bridle. The point P, in CD, is the guided point whose mo- 
tion we have discussed. If we draw AP, and produce it 
until it cuts EF in F, the latter point will have a motion 
similar to P. This will be clear when we consider that in 
all positions EF is parallel to CP; then, since AE and AC 
are fixed lengths, we have for any position two similar tri- 

AP AC 

angles ACP and AEF ; hence = = constant. So 

*^ AF AE 

that, if P describes a straight line, F will also move in a 

straight line parallel to the path of P. 

261. Scott Russell's Motion. — A combination due to 

Mr. Scott Russell, similar to that of Fig. 120, is usually 



AGGREGATE COMBINATIONS. 



283 



classed as an exact straight-line motion. In that figure, if 
the point Q be compelled to move in straight guides along 
AL^ the point Fwill move in a straight path AV^ the arm 
AP oscillating instead of performing complete revolutions. 
This would scarcely seem to be entitled to the term " exact 
motion," since it depends upon the accuracy of the guides 
at Q, the necessity of which it is the object of straight-line 
motions to avoid. 

262. Grasshopper Motion. — A form of the above 
motion in which the guides are replaced by a comparatively 
long radius-rod perpendicular to AL in mid position, and con- 
nected to Q, is approximate, and is known as the "Grass- 
hopper Motion." 




In Fig. 188, let p, P, and x>' be the extreme and middle 
positions of the guided point, lying in one straight line. 
Draw the straight line DFB^ perpendicular to pPp' ; and 
lay off pa = ^a = PA = the proposed length of the guid- 



284 



ELEMENTARY MECHANISM. 



ing bar, so as to find the extreme positions A and a of its 
farther end. This end is to be guided by a lever centred at 
C; that lever being so long as to make the point A describe 
a very flat circular arc, deviating very little from a straight 
line. 

Choose a convenient point b for the attachment of the 
bridle to the bar AB, and lay off pb = j}'b' = PB, so as 
to find the extreme and middle positions of that point. 
Next find the centre Z) of a circular arc passing through 
6, B, and b' ; then D will be the axis of motion of the bridle 
Db. The error of this parallel motion is less, as b is nearer 
the middle of |9a. 

263. Robert's Approximate Straight-Line Motion. 
— Fig. 189 illustrates Robert's parallel motion. Two equal 
arms AG and BD are jointed to fixed centres at one end, 




connected at the other end to the ends of the base of a 
rigid isosceles triangle CPD. In this triangle, CF = DP = 
AC = BD, and CD = ^AB. Tt is evident that in the mid 
position shown, the point P is in the straight line AB ; also, 
that it will lie in this line when PD coincides with BD at one 



AGGREGATE COMBINATIONS. 



285 



end of the stroke, and when PC eoiucides with -4(7 at the 
other end of the stroke. Between these positions, liowever, 
P deviates slightly from AB. 

264. Tchebicheff s Approximate Straight - Line 
Motion. — Another close approximation to a straight-line 
motion is that due to Prof. Tchebicheff of St. Petersburg, 
and illustrated by Fig. 190. The arms are of the following 
proportions : Let AB = 4, then AC = BD = 5, and CD = 2, 




Fig. 19 o 



The path of the guided point P, midway between C and Z>, 
will then closely approximate to a straight line parallel to 
AB. It may be easily proved that the distance of P from 
AB is the same at the ends of the stroke, where P is in 
the perpendiculars to AB through A and 5, and in the mid 
position being that shown in the figure. In intermediate 
positions P deviates slightly from a straight line. Both this 
and the preceding motion give a closer approximation than 
can be obtained by Watt's motion. 

265. A Trammel is a device for drawing ellipses. It 
consists (Fig. 191) of a bar, P(7Z>, carrying a pencil at P, 
and fitted with pins, or pieces mounted on pins, which slide 
in grooves, as shown in the figure. The grooves are usually 
at right angles with each other, and the cross-shaped piece 



286 



ELEMENTARY MECHANISM. 



ill which they are formed is fastened in place on the paper. 
Let PD = a = the semi-major axis of the ellipse to be 
drawn, PC = 6 = the semi-minor axis, PM = x, and P2{ 
= y. Then we have 

^= - = smPDM= sin<^; 

:^=| = COsaPJV^= C0S(5!>. 
PL/ 



+ 



sin^<^ + cos^<^ = 1, 



which is the equation of an ellipse. By varying the lengths 
PC and PD, ellipses of different sizes and eccentricities can 
be drawn. 

B 




266. Oval Clmck If in Fig. 191 we keep the bar 

CPD stationary, and turn the grooved piece and paper, an 
ellipse will be described upon the paper by the point P as 



AGGREGATE COMBINATIONS. 



287 



before. This fact is taken advaotage of in the so-called 
"oval" chuck for turning ellipses, and of which Fig. 192 
illustrates the principle. In this figure P is the cutting tool, 
C the centre of the mandrel of the lathe, and D the centre 
of a circular piece which is fixed to the headstock of the 
lathe. One part of the chuck is fixed to the mandrel, and 
has cut in it a diametral slot represented by aCh. A second 




B'ifif.l9S 



part of the chuck, being that which carries the piece to be 
turned, has two lugs which project through the slot aCh and 
form part of two straight pieces, represented by ad and 6c, 
which slide on the circular piece previously referred to. The 
result is, that, as the mandrel revolves, the piece being turned, 
or the work, receives a combination of this motion of rota- 
tion and a reciprocating motion in the slot, by which the 
distance of the centre of the work from the tool is varied 
in the manner necessary to form an ellipse. Draw De par- 
allel to Co, and CO perpendicular to Ga. Then when the 
work has been turned about C through the angle a' (7a, it 
has also been moved through C the distance OD. AYe now 
see that the triangle COD of Fig. 192 corresponds to COD 



ELEMENTARY MECHANISM. 



of Fig. 191, and drawing PJf perpendicular to De, we have, 

as before, 

PM 

~= sin PDM= sin^; 



^= cosPDM= cos(jf,; 



or is for the instant the centre of the ellipse. Evidently 
since P, (7, and D are fixed, the position of this centre is con- 
stantly changing, lying always at the junction of a perpen- 
dicular to aCb through O, and a parallel to aCb through D. 



APPENDIX. 



61. The Liogaritlimic Spiral. — The rolling properties 
of two equal logarithmic spirals can be readily jDroved from 
the polar equation r — rte"^. From this equation and the 

relation ds = \^dr'^ -\-r''dd'^, we have ;7"-\/l- + ~2 = 

constant. That is^ the rate of increase of the length of the 
curve is proportional to the rate of increase of the radius 
vector. Hence, if two equal logarithmic spirals are placed 
in contact in reversed positions as in Fig. 41, and one is 
rotated about its pole, motion is transmitted to the other 
without sliding, i.e., the contact is pure rolling contact. 

62. To Construct the Logarithmic Spiral.— Hav- 
ing given two points on the curve such as A and D, Fig. 
40, a third point such as B may be found as follows : The 
angle ADD equals the angle BOD, and OD is a mean pro- 
portional between OA and OB. Therefore, lay off 0E= 
OD perpendicular to OA. As the angle AEC is necessarily 
a right angle, draw a perpendicular, using triangles, to AE 
through E. The intersection of this perpendicular with 
AO, produced, at G gives 0G= OB, the radius required. 
Points on the curve having radii greater than OA can be 
similarly found. 

68. Interchangeable Lohed Wheels.— The mathe- 
matical proof of the rolling properties of interchangeable 
lobed wheels constructed by this method, as developed by 
Prof. H. B. Gale, is to be found in the Journal of the 
Franklin Institute, for February, 1891. 



200 



ELT::\rT:XTA'RT :vrT!CITAXISM. 



144. Skew Bevel Wheels. — For a very complete dis- 
cussion of Skew Bevel Wheels^ and Twisted or Spiral Gear- 
ing, see articles by Mr. George B. Grant and others in 
the American Macliinist for May 19th, 1888; Sept. 5th and 
Oct. 10th, 1889; Jnly 31st, Ang. 7th, Aug. 28th, Nov. 13th, 
Dec. 18th, and Dec. 25th, 1890. 

154. Cams. — Fig. 193 illustrates the application of the 
principle of parallel curves as referred to in Articles 122 
and 151 in deriving the practical cam curve from the theo- 
retical cam curve, or pitch line, which would transmit the 
desired motion to a jooint. 

To find the actual shape of the practical cam, let the 
full line in Fig. 193 be the pitch line, and let Pa be a con- 
venient radius for the roller. Then, with a radius equal to 
Pa, and with centres on the pitch 
lines small distances apart, de- 
scribe arcs toward the centre of 
the cam, as shown in Fig. 193. 
These small arcs evidently repre- 
sent successive positions of the 
roller as compared with the cam. 
For convenience we suppose the 
roller to move around the cam 
instead of moving the latter under 
the roller. If we now draw a 
curve which just touches these 
small arcs, or is tangent to them, 
it will be the curve according to which the actual cam 
should be made in order to produce very nearly the same 
motion, by means of the roller, as would be produced by 
the revolution of the pitcli line under the point. The use 
of a roller is, however, apt to introduce errors. For exam- 
ple, suppose that the pitch line forms a point as at ^: then 
it is clear that the point d, at which the two sides of the 
actual cam meet, is at a greater distance from B than the 




APPENDIX. ^91 

length of the radius Pa. Therefore the cam as made would 
not lift the roller as far as the pitch line would lift the point, 
by the difference between Bd and Pa. It follows that the 
smaller the roller is made the more nearly the motion pro- 
duced by the actual cam will agree with that of the pitch 
line and point. On the other hand, by attempting to use 
too large a roller, the motion which would be obtained may 
differ considerably from that of the pitch line. For ex- 
ample, in Fig. 193, suppose Pf to be the radius of the 
roller. Then proceeding by drawing arcs, as before, we 
find that they overlap so that the derived curve for the 
cam would have a corner at g, and that from f to g the 
motion would be very different from that required, as 
shown by the pitch line. This is, of course, an extreme 
case, and is given simply to illustrate the principle. Whether 
or not the size of roller selected in any case in practice is 
too great, can be very readily established by making a draw- 
ing to a large scale, and drawing a sufficient number of arcs 
to represent the successive positions of the roller. 

It is sometimes desirable, when a cam is to drive in both 
directions, that it should work between two rollers, and be 
Iways in contact with both 



en 



of them. Fig. 194 shows the 
pitch line of a cam to work in 
this manner. The condition is 
that the distance between the 
two edges of the cam, meas- 
ured across the centre, must 
be constant and equal to the 
distance between the centres 
of the rollers. Let ahcdefg 
be the curve as laid out for irigrio^ 

tlie pitch line for the for- 
ward motion. Then aCg must be the distance between 
the centres of the rollers. To find the radius Cb' lay off 




292 



ELEMENTARY MECHA1S"ISM. 



on aCg the distance ah = Cb, then Jig = ag — Ch is the 
length of CJ)^, to be laid off from C» In the same manner 
we lay of^ ah = Cc and CV = hg, and so on, thus finding 
t]ie points V , c', d', e', and/' on the curve for the back- 
ward motion. It follows from the construction that the 
distances ag,ff, ee\ etc., are all equal, and that the curve 
as laid out for the forward motion controls the backward 
motion. 

We will now examine the form of cam motion in which 
the cam acts upon a flat surface, such as the face of a 
^Mifting toe," or that of a yoke which rests upon the cam. 
In Eig. 195 let C be the centre about which a cam is to 




Fig. 195 

turn through a half -revolution and lift a piece B from a to 
the positions 1, 2, 3, and 4 while the cam turns through 
the arcs ah, he, cd, and de. When the cam has turned 
through the arc aJ), Cb will be in the vertical centre line 
€%, and the face of the piece B will be in the position in- 
dicated by the line drawn through 1 perpendicularly to 



APPENDIX. 293 

the centre line. If the cam is made so that it just touches 
the line Ijy at any point, or is tangent to it, the desired re- 
sult will be obtained. Therefore if we lay ofl Cf = Cl and 
draw/6 perpendicular to Cf, the only essential condition, 
so far as this position is concerned, is that the cam curve 
shall touch /6 at some point, such as 6. 

Another condition is that, in the position in which the 
cam is drawn, the curve must not rise above the horizontal 
line through a, or the tangent to the base circle at that 
point, since in its lowest position B rests upon the cam 
at a. 

Proceeding to the successive positions 2, 3, and 4, we lay 
off Cg — 02, Ch = C3, Ck = (74, and draw perpendiculars 
to the radial lines at g, //, and Ic; we can then complete the 
cam curve by drawing it tangent to these last lines. In the 
figure it has been further assumed that, after the half-rev- 
olution of the cam, the point of contact between the cam 
and piece B is to be on the centre line, therefore ^ is a 
point on the curve. An examination of the figure will 
make it clear that, according to this construction, the 
point of contact moves from the centre line out along the 
face of B until it is at a distance f6 to the right of the 
centre line, when Cb arrives at Ca. It then moves back 
toward the centre line, since, as drawn, g7 is less than /6 
and h8 is less than ^7 until, on the completion of the half- 
revolution, it is again on the centre line at k. The neces- 
sary length of bearing surface on B is therefore equal to 

When a cam of the form shown in Fig. 195 is to make 
complete revolutions and drive in both directions, it may 
be enclosed in a yoke, of which the two working faces are 
the distance ak apart. To complete this cam to work in 
such a yoke, we proceed in much the same manner as for 
a cam which is to work between two rollers. Lay off Cl = 
ok — Cf, Cm = ale — Cg, etc., and draw perpendiculars to 



294 



ELEMENTARY MECHAKISM. 



these lines as ??9, mlO, and 111. If we now complete the 
cam by drawing a curve tangent to these last-mentioned 
lines, it will work in the yoke as required, since the dis- 
tance between parallel tangents to the curve, such as n^ 
and JiS, mlO and g'7, is constant and equal to ak. As 
has been before stated, to secui"e satisfactory results the 
drawing should be to a large scale, and many points found. 
In a construction such as just described there is, of 
course, considerable sliding. This can be reduced to a 
small amount by connecting the cam to the driving 
mechanism so that it shall vibrate through a small angle. 
For example, in Fig. 196 the cam is to turn through the 




^ ITigJ 196 

arc ah of the base circle, and lift the piece B from a to c. 
The method of construction is the same as in Fig. 195, and 
is clear from the figure. It will be noticed that the point 
of contact gradually moves away from the centre line until 
it is at a distance equal to de at the end of the motion. 
The length of face of B or mn should therefore be equal 
to cle. By reducing the angle through which the cam is 
intended to vibrate still further, the curve can be made 



ArPEisT)ix. 295 

still flatter, and therefore more nearly equal in length to 
the face of the lifting toe, the amount of sliding being cor- 
respondingly decreased. This form of cam motion will be 
recognized as that used in Stevens^ cut-off motion. 

164. Pin and Slotted Crank.— In Fig. 197 is illus- 
trated the special case of the quick-return motion shown 




Fig. 107 

in Fig. 106, in which AB exceeds F ; in other words, the 
centre B lies outside of the path of the pin F so that the 
arm BP does not revolve but only oscillates. If the 
arm AF revolves at a constant speed, the periods of the 
two strokes are in the ratio of the arcs FEF' and F'DP, 
If AF be shortened to Aj:), the travel of C is reduced from 
CC to cc' y and the periods are in the ratio of the arcs j!?^^' 
and li'd'p. 

218. Cone Pulleys. — A method of determining the 
diameters of cone or step pulleys which will work satisfac- 
torily together when connected by an open belt has been 
developed by Mr. 0. A. Smith and is to be found in detail 
in Vol. X, Transactions of the American Society of Me- 
chanical Engineers. The graphical construction when the 



296 



ELTi^klEXTAEY MECHANISM. 



greatest belt angle does not exceed 18" is as follows : In 
Fig. 108 lay off the distance between shaft centres ^2^ and 
draw the circles D^ and d^ , equal to the first pair of pul- 
leys which are previously determined by known conditions. 




Draw ML tangent to the circles D^ and d^. From the 
point B, midway between E and F, erect BG perpen- 
dicular to EF and make BG = .3UEF. With G^ as a 
centre, draw a circle tangent to ML. Then the belt line 
of any other pair of pulleys must be tangent to the circle 
G, as indicated in Fig. 198. Thus to find the proper size 
of pulley to work with any other pulley d^, draw HI tan- 
gent to circle d^ and also tangent to circle described about 
G; then a circle I)^ drawn tangent to HI will be the size 
required. 



-^^. 



PROBLEMS. 



1. An ecgiue makes 600 strokes per minute. Fly-wheel is on the 
crank shaft. Find the linear and angular velocit}' of a point in the 
fly-wheel 3 feet from the centre of the shaft. 

A71S. a = 1884.96 ; V = 5654.88 feet per minute. 

2. The speed of the periphery of a wheel 8 feet in diameter is 
4,000 feet per minute. Find the linear velocity of a point 3| feet 
from the centre. 

3. A point in a fly-wheel, 4 feet from the centre of the wheel, 
moves through 2,500 feet per minute. The stroke of the engine 
being 2 feet, find the mean piston speed. 

Ans. V = 397.89 feet per minute. 

if^ A locomotive moving at the rate of 35 miles per hour has 
driving wheels 63 inches in diameter and cylinders 24 inches siroke. 
Find the linear and angular velocities of the crank-pins relatively to 
the frame of the engine. 

5. Two shafts are centred 4 feet apart. Find the diameters of 
wheels to work by rolling contact, so that the driving shaft will 
make 5 revolutions while the following shaft makes 7 revolutions. 

Ans. Driver, 28 inches ; follower, 20 inches. 

6. The distance between the centres of two shafts = 54 inches. 
The driving-shaft makes 80 revolutions per minute. The follower 
is to make 100 revolutions per minute. Find the diameter of wheels 
for rolling contact. 

7. A shaft making 120 revolutions per minute is to drive by spur 
gearing a second shaft 28 inches from it at a speed of 300 revolutions 
per minute. Find diameters of pitch circles. 

297 



29S ELEMEKTARY MECHAKISM. 

3 a' 

8. Velocity ratio to be transmitted = - = — . Diameter of the 

driver is 15 iuclies. Find the diameter of the follower, and the dis- 
tance between paruilel axes. (Direct contact.) 

Ails. Diameter, 20 inches , distance, 17 1 inches. 

9. A wheel 32 inches in diameter is fixed on a shaft making 325 
revolutious in 5 minutes. This wheel and shaft are to drive a second 
wheel by rolling contact, so that the latter will make 52 revolutions 
per minute. Find the size of the second wheel, and the distance 
between the centres of the wheels. 



10. Given two intersecting axes at right angles, velocity ratio 

4 

— . Show how to find the pitch cones graphically. 

o 



a 4 
a 



11. The angle between two intersecting axes is 75°. Show how 
to find graphically the sizes and positions of conical frusta which 

.... «' 65 

Will transmit a velocity ratio — = -— . 

12. P = circular pitch, N = number of teeth. 
B = pitch diameter, if = diametral pitch. 

(1) Given P = 2^ inches, N= 40. Find B. 

(2) Given P = 1^ inches, N= 75. Find B. 

(3) Given P— f inch, B = 12 inches. Find iV. 

(4) Given B = 24 inches, Ii= 50. Find P. 

(5) Given 8-pitch wheel, K=40. Find Z>. 

(6) Given 3-pitch wheel, W= 60. Find B. 

(7) Given 4-pitch wheel, B = 20 inches. Find iV. 

(8) Given 2-pitch wheel, Z) = 35 inches. Find If. 

(9) Given D = 15 inches, JV = 75. Find M. 
(10) Given B = 21 inches, W= 81. Find M. 

13. Two axes 27 inches apart are to be connected by two 2-pitch 
wheels. Velocity ratio |. Find diameters of pitch circles and 
numbers of teeth. Ans. Numbers of teeth, 63 and 45. 

14. Prove that two equal circles set equally eccentric will not 

roll together. 

15. Two spur wheels in gear have 80 and 30 teeth, respectively, 
cycloidal system, and 1^ inches circular pitch. What is the correct 
distance between centres of shafts? 



PROBLEMS. 299 

16. GiveQ the angle between two intersecting axes = 60**, con- 

, . . . «' 3 

struct cones to give a velocity ratio of — = — . 

IT. The distance between centres of two parallel shafts is 20 

inches. They are connected by two 3-pitch spur wheels such that 

a' 3 

— = — . What are the numbers of teeth ? 

a 5 

18. Construct three teeth on each of a pair of 4-pitch cycloidal 
spur gears, showing points of coming in contact and quitting contact, 

having given : Diameters of describing circles = — of the pitch 

diameters ; addendum = one pitch part ; distance between wheel 

a 1 
centres = 6 inches ; — = — . 

19. Show by construction whether or not two 8-leaved pinions 
having radial flanks, epicycloidal faces and arc of recess = f pitch 
will work together. 

20. Construct a cam curve as follows : Diameter of base circle 
= 3 inches ; line of motion of driven point is vertical and passes 
i inch to the right of centre of circle; stroke of point = 2 inches ; 
point is to rise with uniform velocity during ^ of a revolution, 
remain stationary \, and descend with uniform velocity during the 
remainder of the revolution. 

21. Construct a cam on a base circle of 3 inches diameter, to 
revolve once per minute, and give to a bar, whose line of motion 
passes through the centre of motion of the cam, a stroke of 2 inches. 
The bar rises during 25 seconds with a uniform velocity ; remains 
at rest 20 seconds ; and descends during the remainder of the revo- 
lution with a uniformly accelerated velocity. 

22. Draw a cam which, by oscillating through an angle of 60°, 
shall give a uniform ascending and descending motion to a bar 
whose line of motion passes 4 inches to right of the centre of the 
cam. Stroke of the bar, 3 inches, 

23. Design a cam on a base circle of 3 inches diameter, to raise a 
point whose line of motion passes one inch to the right of the centre 
of motion of the cam, by a uniform step-by-step motion, during 
f of a revolution of the cam, and allow it to descend with uniform 
velocity during the remaining i of the revolution. 



300 ELEMEI^TARY MECHA>^ISM. 

24. In Fig. 106, given AB = S inches, AP = 5 inches ; find 
length and position of the slotted arm when — = 1. 

25. In Fig. 107, given AP = 2 feet, AB = 1 foot, BG = dh feet, 
CQ — 6 feet. AB is vertical, and aQ is horizontal. P revolves in 
the direction of arrow, making one revolution per minute. 

(1) Find length of stroke of Q, '] 

(2) Find time of forward stroke in seconds, y by computation. 

(3) Find time of backward stioke in seconds, J 

(4) Find position of P when Q is at the middle of ^ 

forward stroke, I 

(5) Find position of P when Q is at the middle of [^ graphically. 

backward stroke, j 

26. Design a quick-return motion such that the periods shall be 
as 7 to 5, and the stroke of the slide = 4 inches. 

27. Design a WhitwortJi quick-return motion to have periods as 
2 to 3, and stroke of tool from 2 inches to 4 inches. 

2§. Construct the curve for a cam on a base circle 3 inches in 
diameter, which by revolving uniformly will give harmonic motion 
to a bar of which the line of motion is vertical and passes f inch to 
the left of the centre. 

29. Having a crank 2 feet long and a connecting-rod 8 feet long, 
find the angle of the crank with line of centres when the piston is at 
the middle of its stroke. 

Ans. ± 82° 49' 9". 

30. Having a crank 1 foot long and a connecting-rod 5 feet long, 
revolutions per minute 120, find piston velocity in feet per minute 
when the crank makes an angle of 45° with the line of centres. 

Ans. 609.22 feet per minute. 

31. Having an engine of 5 feet stroke and a 10-foot connecting- 
rod, find distance of the piston from the end of stroke when the 
crank has made J of a revolution. 

Ans. 2 feet 2.19 inches. 

32. Having an engine of 3 feet stroke, connecting-rod 10.} feet 
long, find what angles the crank makes with line of centres when 
the velocity of the piston equals that of the crank. 

Ans. Sin-^ X. 



PROBLEMS. 301 

33. Given the stroke of an engine = 6 feet and length of con- 
necting-rod = 12 feet; find the distance of the piston from the end 
of the stroke when the crank has turned through 135^ from the 
head end. 

34. Given the length of a crank :inn = 20 inches, connecting-rod 
= 70 inches, distance between line of motion of cross-head and 
shaft centre = 30 inches ; find the length of the stroke and the rela- 
tive peiiods of the two strokes. 

35. Given a rotating arm 2 feet long, an oscillating arm 3 feet 
long, distance between the centres = 5 feet, and length of link = 4 
feet ; will this motion work satisfactorily or not, and why? 

36. Having a beam engine of 10 feet stroke, 13 feet between the 
centres of beam and cylinder, find the best length for the beam arm. 

37. In a beam engine, having given the perpendicular distance 
between the centre line of the cylinder and the beam bearings = 7 
feet, and the stroke = 5 feet, find the best length for the beam arm. 

3§. Given a rocker arm which vibrates through 45° each side of 
its mid position ; stroke of follower = 20 iuches ; find length of 
rocker arm which will give the minimum vibration to the follower. 

39. Show graphically how to construct a quick-return motion by 

, ,. , , , period of advance 3 

lomted hnks, such that -. — -. — ^^ = — . 

•' period of return 2 

40. Design a cam on a base circle of 2 inches diameter, to give to 
a point whose line of motion passes i inch to the right of the centre 
of motion of the cam, the same motion as piston in problem 31. 

a' 1 

41. Connect two parallel shafts by a crossed belt, so that — = ^t 

and find the length of the belt by exact calculation. 

42. Two shafts are to be connected by an open belt ; distance be- 

ty' 2 
tween axes = 10 feet and — = — . Find diameters of pulleys and 
<x o 

the length of the belt. 

43. A shaft distant 30 feet from a main shaft and parallel to it is 
to be driven at a speed of 80 revolutions per minute by a single belt 
2^ inches wide. Revolutions of main shaft, 248 per minute. Select 
pulleys from a manufacturer's catalogue to obtain the desired speed 
as accurately as practicable, using an intermediate shaft if advisable. 



302 ELEMENTARY MECHANISM. 

44. A countershaft 7 feet from a main shaft which revolves 240 
times per minute is to be driven at speeds of 80, 120, 240, 360, and 
480 revolutions per minute. Find diameters for cone pulleys to 
work with an open belt, the smallest pulley to be not less than 5 
inches in diameter. 

45. A pulley (A) on a driving-shaft drives p\illey (B) by a crossed 
belt. A spur gear (C) on shaft with (B) drives pinion (I)). Pulley 
(B), on the shaft with (D), drives pulley (F) by an open belt. 

Given J. = 20 inches diameter, 40 revolutions per minute. 
Given B = 15 inches diameter. 
Given G = 90 teeth, I) = 15 teeth. 
Given E = dO inches diameter, F = 10 inch'es diameter. 
Find number of revolutions per minute of F, and direction of 
rotation relatively to A. 

46. An engine of 3 feet stroke, piston speed of 360 feet per min- 
ute, has a main driving-pulley 8 feet in diameter, from which is 
driven a pulley 4 feet in diameter. A pump having a plunger dis- 
placement of 2 gallons is to be driven from a shaft carrying the 4- 
foot pulley, and is to pump 5,000 gallons per hour. Find arrange- 
ment of the connecting train of mechanism, by belts or gearing. 

47. A lathe has a set of change wheels whose pitch diameters are 
2 inches, 3 inches, 5 inches, 6 inches, 7| inches, and 9 inches re- 
spectively. Leading screw has 4 threads to the inch and is right- 
handed. Distance between the centres of spindle and leading screw 
is 16 inches. Select and arrange wheels to cut a left-handed screw 
of 6 threads to the inch. 

48. Given e = 1250, w = 125, and p = 15. Find numbers for 
teeth for a train of spur gears. 

49. Find trains for an 8-day clock, 16 turns of weight cord on 
barrel. The escape wheel has 30 teeth ; number of teeth on wheels 
not to exceed 96 ; number of teeth on pinions not less than 8. Re- 
quired hour, minute, and seconds hands. 

50. Find the trains for a 32-day clock, the barrel to carry 24 coils 
of the weight cord ; pinions to have not less than 8, and the wlieels 
not over 108 teeth ; swing wheel (escape wheel) to have 60 teeth, and 
the pendulum to make 120 vibrations per minute. Required hour, 
minute, and seconds hands. 



PKOBLEMS. o03 

51. Find trains for a 12-day clock ; 18 turns of weight cord on 
barrel ; escape wheel revolves twice per minute ; pendulum makes 
120 beats per minute ; least number of teeth for pinions = 9 ; 
greatest number for wheels = 108. Required hour, minute, and 
seconds hands. 

52. Find trains for an 8-day clock. Pendulum makes 150 vibra- 
tions per minute ; swing wheel has 25 teeth ; dead-beat escapement ; 
least number of teeth for pinions = 10 ; greatest number of wheels 



53. A lathe has 4 threads per inch on a right-handed leading 
screw. Find the sizes of least number of change wheels to cut 
right-handed threads of 5, 6, 8, 9, and 10 to the inch. Smallest 
wheel to have 20 teeth. Arrange table for change wheels for the 
various cuts. 

54. The leading screw of a lathe has 8 threads per inch. The 
change wheels have 20, 24, 30, 36, 40, 42, 48, 50, 64, 72, 78, 84, 90, 
96, and 100 teeth. The wheels are compounded. Select wheels to 
cut a triple-thread screw having square threads f inch square. 

55. The leading screw of a lathe is to have 6 threads per inch. 
Find numbers of teeth for change wheels to cut all U. S. standard 
bolt and pipe threads up to 6 inches, and select Brown and Sharpe 
involute cutters to cut the gears. 

56. A countershaft to make 250 revolutions per minute is to be 
erected in a square- cornered room, 3 feet from the north wall and 3 
feet from the ceiling, and is to be driven from a shaft which cuts 
across the northwest corner in a plane parallel to the plane of the 
ceiling and 8 feet from it, intersecting the west wall at 6 feet from 
the corner and the north wall at 4 feet from the corner. The sj^ei'd 
of this shaft is 40 revolutions per minute. Design a method of con- 
nection to accomplish the desired result. 

57. Find numbers of teeth for a train to give approximately e = 
194 with an error of less than 1 ; maximum number of teeth for 
wheel = 90 ; minimum number for pinion = 12. 

58. Find numbers for a train of wheels to transmit a velocity 
ratio of 853 with an error of not more than 1 in 150,000. 



304: ELEMENTARY MECHANISM. 

59. Find a train as in problem 58 to give e = 251, with error not 
more than 1 in 8,700. 

60. Design a drill press (Fig. 174), pitch of screw to be | inch ; 
drill to make 60 revohilions per minute ; driving axis to make 40 
revolutions per minute. Drill to descend gV i"ch per levolution. 

61. In Fig. 176, Chas 121 teeth and is fixed, I) has 120 teeth, d 
has 119 teeth, E= 120 teeth. Find how many revolutions of arm A 
will cause E to revolve once. 

62. In Fig. 176, (7 is a fixed wheel and has 20 teeth, i) = 36 teeth, 
(f = 24 teeth, E= d2 teeth. Find velocity ratio = ■^. 

63. In Fig. 179, G has 30 teeth and ^ has 40 teeth. Required 

G 
velocity ratio = jry:. 

64. In Fig. 179, G makes 8 turns while OD makes 5 turns. Find 
number of teeth for ^ and C. 

65. In an epicyclic train such as Fig. 176, the numbers of teeth 
are : C = 60, B = 50, d = 54, and E = 45. Revolutions of C= 40 
per minute from right to left. Revolutions of the arm = 50 per 
minute from left to right. What will be the exact number of revo- 
lutions per minute of E? 

66. Find numbers for a four-wheel epicyclic train such that when 
the arm revolves 400 times the last wheel will revolve once. 

67. Design an epicyclic train of spur gears by which the velocity 
ratio — = 370 can be exactly transmitted, by wheels having not 
over 45 teeth. 

68. In Fig. 182, let the arm BG' be suppressed, and let P' be 
guided in a circle drawn on AP as a diameter. Prove that G will 
move in a straight line perpendicular to AP. 

69. Find dimensions of a trammel to describe an ellipse of which 
the major axis is 3^ times as long as the minor axis. 



INDEX. 



PAGE 

Action, angle and arc of 86 

Action, line of 26 

Action of bevel and spur wheels, rela- 
tive 156 

Addendum 85 

Addendum circle 85 

Aggregate combinations 263 

Aggregate velocities by linkwork . . 263 
Analogy between cones and hyperbo- 

loids 42 

Anchor escapement 188 

Angle of action, approach and recess, 86 

Angular velocity 4 

Annular wheels 47, 101, 138 

Approach, angle and arc of .... 86 
Approximate forms of teeth .... 113 
Approximate numbers for trains . . 260 
Approximate straight-line motion, 

Roberts' 284 

Approximate straight-line motion, 

Tchebicheff's 285 

Approximate straight-line motion. 

Watt's 278 

Arc of action, approach and recess . 86 
Arc of action and pitch, relation 

between 95 

Arcs, rectification of circular ... 74 

Axes, least number of 250 

Automatic drill-feed 267 

Bands 15 

Backlash 86 

Bar and cam, slotted 183 

Bar and pin, slotted 182 

-BelUrauk ....--. ^^ . .^ .. ^09 



PAGE 
Belts 228 

Belts, length of 236 

Belts, shifting 231 

Belts, twisted 232 

Bevel gearing 47, 151 

Bevel and spur wheels, relative action 
of 



156 

Bevel wheels 151 

Bevel wheels, skew 157 

Boehm's coupling 208 

Cams 165 

Cam and slotted bar 183 

Cam cui-ve, construction of ... . 166 
Cam for complete revolutions . . . 169 

Catch, frictional 223 

Centre, instantaneous 18 

Chuck, oval 286 

Circle, addendum 85 

" of the gorge 36 

" i^itch 45 

" size of describing 94 

Circular arcs, rectification of ... 74 

" pitch 80 

Clearance 86 

Click and ratchet 219 

*' double-acting 222 

" reversible 220 

" silent 221 

Clock, eight-day 256 

" month 257 

" trains, examples of 251 

Clockwork 245 

Cog, hunting 260 

1 Combination, elementary .- . , . . -7 
3q5 



306 



INDEX. 



PAGE 

Component motions 11 

CompoBition of motions 11 

Condition of constant velocity ratio . 24 
" of rolling contact . . . . 24 
Cones and hyperboloids, analogy be- 
tween 42 

Cones, rolling 29 

Conjugate curves 73 

Connectors, wrapping 15, 227 

Contact between teeth, frequency of . 259 
" condition of rolling .... 24 
" motions, directional relation 

in 24 

" motions, velocity ratio in . . 21 

" rolling 14, 27 

'« sliding 14, 66 

Coupling, Boehm's 208 

'* or universal joint, Hoolie's, 214 

" Oldham's 185 

Crank 193 

Cranks, bell 209 

Crank and pin, slotted 179 

Crank substitute. Watt's 272 

Crown-wheel escapement 187 

Curves, conjugate 73 

Curvilinear motion 2 

Cycle 6 

Cycloid, construction of 79 

Cylinders, rolling 27 

Dead-beat escapement 190 

Derived tooth outline, pin gearing . 133 

Describing circle, size of 94 

Diameter of pin, limiting 136 

" pins of sensible .... 133 

" table of pitch 81 

Diametral pitch 82 

Differential pulley, Weston's . . . 265 

Directional relation 8 

Directional relation in contact mo- 
tions 24 

Directional relation in linkwork . . 20 
Directional relation in trains . . . 243 
Directional relation in wrapping con- 
nectors 21 

Double-acting click 222 

Drag link 204 

Drill feed, automatic 267 

Driver 7 

Driver, least follower for given . , Hi. 



PAGE 

Eccentric 199 

•' rod 193 

Eight-day clock 256 

Elementary combination 7 

Ellipses, compulsory rotation of roll- 
ing 60 

Ellipses, lobed wheels from .... 57 

" rolling 55 

Endless sciew 174 

Epicyclic trains 268 

" " velocity ratio in . . 269 

Epicycloid, construction of .... 76 

" and hypocycloid . ... 67 

** and pin 69 

" and radial line .... 68 

Escapement, anchor ....... 188 

" crown-wheel .... 187 

" dead-beat- 190 

Exact straight-line motion, Peaucel- 

lier's 275 

Exact straight-line motion, Russell's, 282 

Faces and flanks, teeth with both , . 92 

Face gearing 49, 159 

Faces or flanks only, teeth with . . 87 

Face of tooth 85 

Feed, automatic drill 267 

Ferguson's paradox 271 

Flank of tooth 85 

Follower 7 

" for given driver, least . . . 144 
Frequency of contact between teeth . 259 

Friction gearing 44 

Frictional catch 223 

Grearing 45 

bevel 47,151 

" classification of 45 

" face 49,159 

" friction 44 

" pin 131 

" screw . 49, 172 

" skew 48,157 

" spur 46, 85 

" twisted . 50, 149 

Gears, mitre 48 

Gorge, circle of the 36 

Grant's odontograph 124 

Graphic representation of motion . . 10 
GraesboppQi motion 28S 



INDEX. 



307 



PAGE 

Guide pulleys 233 

Hooke's coupling or universal joint . 214 

Hour-glass worm . , 177 

Hunting cog 260 

Hyperboloid of revolution .... 34 

Hyperboloids, rolling 36 

HyperboLoids, velocity ratio of roll- 
ing 37 

Hyperboloid and cones, analogy be- 
tween 42 

Hypocycloid and epicycloid .... 67 
'* construction of .... 78 

Idle wheel 244 

Inside screw 173 

Instantaneous centre 18 

Interchangeable lobed wheels ... 59 
'• spur wheels . . . 97 

Intermittent motion 2, 62 

Involutes 70 

Involute, construction of 79 

'• teeth, interference of . . . 109 
" teeth, peculiar properties 

of Ill 

•' system of teeth 104 

Length of belts 236 

Line of action 26 

" pitch 45 

" epicycloid and radial .... 68 

Linear velocity 3 

Links 15,193 

Linkwork 193 

" aggregate velocities by . . 263 
" directional relation in . . 20 
•* multiplication of oscillation 

by .211 

** quick return motion by . . 213 
" rapidly varying velocity by, 212 
" circular motion into recti- 
linear reciprocation and 
the reverse by .... 194 
" continuous rotation into ro- 
tative reciprocation and 
the reverse by .... 200 
•* transmission of continuous 

rotation by ..... 204 

" velocity ratio in .... 16 

Lobed wheels 56 



PAGE 

Low-numbered pinions 140 

Logarithmic spiral 52 

" " construction of . . 53 

'• " lobed wheels from, 56 

Machine, definition of 1 

Machines, parts of 6 

Mangle rack 65 

" wheels 63 

Mechanism, definition of pure ... 1 

'« train of 7,240 

Mitre geai's 48 

Month clock 257 

Motion and rest 2 

Motions, composition of 11 

Motion, curvilinear 2 

directional relation in contact, 24 
graphic representation of . . 10 

grasshopper 283 

intermittent 2, 62 

kinds of 2 

modes of transmission of, 7, 16 

oscillating 2 

parallel 275 

parallelogram of 11 

Peaucellier's exact straight- 
line 275 

periodic 5 

polygon of 12 

quick return by linkwork . . 213 

reciprocating 2 

rectilinear 2 

resolution of ....... 13 

resultant 11 

Roberts' approximate straight- 
line 284 

Russell's exact straight-line . 282 
screw for variable .... 178 
Tchebicheff's approximate 

straight-line 285 

velocity ratio in contact . . 21 

vibrating 2 

Watt's approximate straight- 
line 278 

Watt's sun and planet . . . 272 
Whitworth's quick return . 181 

Non-circular wheels 150 

Notation 247 

Numbers for trains, approximate . . 260 



308 



INDEX. 



PAGE 

Odontograph, Grant's 124 

" improved Willis's . . 122 

" Robinson's templet . 126 

" Willis's 119 

Oldham's coupling 185 

Oscillating motion 2 

Oscillations by linkwork, multiplica- 
tions of 211 

Outside screw 173 

Oval chuck 286 

Paradox, Ferguson's 271 

Parallel motions 275 

Parallelogram of motions 11 

Parts of machines 6 

Path 2 

Peaucellier's exact straight-line mo- 
tion .... ... 275 

Period 6 

Periodic motion 5 

Pin and epicycloid 69 

Pin and slotted bar 182 

Pin and slotted crank ...... 179 

Pin gearing .......... 131 

Pin gearing, limiting diameter of 

pins 136 

Pin gearing, pins of sensible diame- 
ter 133 

Pinions . . . c 47 

-" low-numbered 140 

" two-leaved 142 

Pitch and arcs of action, relation be- 
tween 95 

Pitch circle 45 

" circular 80 

•' diameters, table of 81 

" diametral 82 

" line 45 

" of screw 172 

" surface . 45 

" point 85 

Polygon of motions 12 

Problems 289 

Properties of involute teeth, pecu- 
liar Ill 

Pulley, convexity of 232 

" guide 233 

" stepped 229 

♦* tightening . . . . •. . .231 
«• Weston's differential ... 265 



PAGE 

Quick return motion by linkwork . . 213 
" " " WMtworth's . 18 

Rack 47 

Rack and wheel 98, 110, 137 

Ratchet, click and . 219 

Ratio, velocity 8 

Recess, angle and arc of ..... 86 

Reciprocating motion 2 

Rectification of circular arcs ... 74 
Rectilinear motion ....... 2 

Relation, directional 8 

Representation of motion, graphic . 10 
Resolution of motion ...... 13 

Rest and motion 2 

Resultant motion 11 

Reversible click 220 

Revolution 2 

" cam for complete .... 169 

" hyperboloid of 34 

Roberts' approximate straight-line 

motion 284 

Robinson's templet odontograph . . 126 

Rolling cones 29 

" contact ........ 14 

" " condition of .... 24 

'• cylinders 27 

" ellipses 55 

" ellipses, compulsory rotation 

of 60 

" hyperboloids 36 

" hyperboloids,velocity ratio of, 37 

Rotation 2 

" synchronal 28 

Russell's exact straight-line motion . 282 

Screw 172 

" endless 174 

" for variable motion .... 178 

" gearing 49,172 

•' inside 173 

" outside 173 

" pitch of 172 

Shifting belts 231 

Similarity in all modes of trans- 
mission 26 

Size of describing circle 94 

Skew bevel wheels 157 

Skew gearing 48 

Sliding contact « , 14 



INDEX. 



309 



PAGE 

Sliding in contact motions, percent- 
age of 40 

Sliding in contact motions, velocity of, 24 

Slotted bar and cam 183 

" bar and pin . . . . . . .182 

" crank and pin 179 

Spiral, logaritlimic 52 

Spiral, logarithmic, lobed wheels 

from 56 

Spur and bevel wheels, relative action 

of 156 

Spur gearing 46, 85 

Stepped pulley 229 

" wheel 50 

Straight-line motion, Peaucellier's 

exact 275 

Straight-line motion, Roberts' approx- 
imate 284 

Straight-line motion, Russell's exact . 282 
Straight-line motion, Tchebicheff's 

approximate 285 

Straight-line motion, Watt's approxi- 
mate 278 

Surface, pitch 45 

Sun and planet motion, Watt's . . . 272 
Synchronal rotations 28 

Tappets 165 

Tchebicheff's approximate straight- 
line motion 285 

Teeth, approximate forms of . . . 113 
" customary dimensions of . . 102 
" definitions of parts of ... 85 
" frequency of contact between, 259 
" interference of involute . . . 109 
" peculiar properties of involute, 111 

" unsymmetrical 148 

" with both faces and flanks . . 92 

" with faces or flanks only . . 87 

Templet odontograph, Robinson's . 126 

Tightening pulleys 231 

Tooth outline in pin gearing, derived, 133 

" " in bevel gearing . . , 153 

Trains, approximate numbers for . . 260 

" directional relation in . . . 243 

" epicyclic .268 

" examples of clock .... 251 
" of mechanism, definition . . 7 
" for rope-making machinery . 273 
*• method of designing . . , 249 



PAGE 

Trains, value of 240 

" velocity ratio in epicyclic . . 269 

Trammel 285 

Transmission of motion, modes of, 7, 16 
Transmission, similarity in all modes 

of 26 

Tredgold's method for bevel wheels . 154 

Twisted belt 232 

" gearing 50, 149 

Two-leaved pinion 142 

Uniform periodic motion 6 

•* velocity 3 

Unsymmetrical teeth 148 

Value of a train . 240 

Variable velocity 4 

Velocity, angular 4 

" constant or uniform ... 3 

" linear 3 

*' of sliding in contact motions, 24 

" ratio. . 8 

" " condition of constant . 24 
" "in contact motions . . 21 
" "in epicyclic trains . . 269 
" " in linkwork .... 16 
" " in wrapping connect- 
ors 20 

" "of rolling hyperbo- 

loids 37 

" " variable 4 

Verge 187 

Vibrating motion 2 

Watt's approximate straight-line mo- 
tion 278 

Watt's crank substitute, or sun and 

planet motion 272 

Weston's differential pulley .... 265 

Wheel and rack 98, 110, 137 

Wheels, annular 47, 101, 138 



" bevel 


. . 151 


'< idle 


. . 244 


" interchangeable . . . 


. . 97 


lobed . 


. . 59 


" lobed . . ^ 


, . 56 


" mangle 


. . 63 


" non-circular . ... 


. . 150 


" skew bevel 


. . 157 


" stepped 


. . 50 



310 



INDEX. 



PAGE 

Wheel, worm and 174 

Whitworth's quick return motion . . 181 
Willis' method of approximation by 

circular arcs 114 

Willis' odontograph 119 

" •• improved . . .122 
Wipers 165 



PAGE 

Worm and wheel 174 

*' hour-glass 177 

Wrapping-connectors .... 15, 227 
Wrapping connectors, directional re- 
lation in 21 

Wrapping connectors, velocity ratio 
in 20 



Deacidified using the Bookkeeper process 
Neutralizing agent: Magnesium Oxide 
Treatment Date: July 2004 

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